Fast compiled graphs¶
This is a Cython implementation of the base class for sparse and dense graphs in Sage. It is not intended for use on its own. Specific graph types should extend this base class and implement missing functionalities. Whenever possible, specific methods should also be overridden with implementations that suit the graph type under consideration.
For an overview of graph data structures in sage, see
overview.
Data structure¶
The class CGraph maintains the following variables:
- cdef int num_verts
- cdef int num_arcs
- cdef int *in_degrees
- cdef int *out_degrees
- cdef bitset_t active_vertices
The bitset active_vertices is a list of all available vertices for use, but
only the ones which are set are considered to actually be in the graph. The
variables num_verts and num_arcs are self-explanatory. Note that
num_verts is the number of bits set in active_vertices, not the full
length of the bitset. The arrays in_degrees and out_degrees are of the
same length as the bitset.
For more information about active vertices, see the documentation for the
method realloc.
- class sage.graphs.base.c_graph.CGraph[source]¶
- Bases: - object- Compiled sparse and dense graphs. - add_arc(u, v)[source]¶
- Add arc - (u, v)to the graph.- INPUT: - u,- v– nonnegative integers; must be in self
 - EXAMPLES: - On the - CGraphlevel, this always produces an error, as there are no vertices:- sage: from sage.graphs.base.c_graph import CGraph sage: G = CGraph() sage: G.add_arc(0, 1) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraph >>> G = CGraph() >>> G.add_arc(Integer(0), Integer(1)) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - It works, once there are vertices and - add_arc_unsafe()is implemented:- sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(5) sage: G.add_arc(0, 1) sage: G.add_arc(4, 7) Traceback (most recent call last): ... LookupError: vertex (7) is not a vertex of the graph sage: G.has_arc(1, 0) False sage: G.has_arc(0, 1) True sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc(0,1) sage: G.add_arc(4,7) Traceback (most recent call last): ... LookupError: vertex (7) is not a vertex of the graph sage: G.has_arc(1,0) False sage: G.has_arc(0,1) True - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(5)) >>> G.add_arc(Integer(0), Integer(1)) >>> G.add_arc(Integer(4), Integer(7)) Traceback (most recent call last): ... LookupError: vertex (7) is not a vertex of the graph >>> G.has_arc(Integer(1), Integer(0)) False >>> G.has_arc(Integer(0), Integer(1)) True >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc(Integer(0),Integer(1)) >>> G.add_arc(Integer(4),Integer(7)) Traceback (most recent call last): ... LookupError: vertex (7) is not a vertex of the graph >>> G.has_arc(Integer(1),Integer(0)) False >>> G.has_arc(Integer(0),Integer(1)) True 
 - add_vertex(k=-1)[source]¶
- Add vertex - kto the graph.- INPUT: - k– nonnegative integer or- -1(default:- -1); if \(k = -1\), a new vertex is added and the integer used is returned. That is, for \(k = -1\), this function will find the first available vertex that is not in- selfand add that vertex to this graph.
 - OUTPUT: - -1– indicates that no vertex was added because the current allocation is already full or the vertex is out of range
- nonnegative integer – this vertex is now guaranteed to be in the graph 
 - See also - add_vertex_unsafe– add a vertex to a graph. This method is potentially unsafe. You should instead use- add_vertex()
- add_vertices– add a bunch of vertices to a graph
 - EXAMPLES: - Adding vertices to a sparse graph: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(3, extra_vertices=3) sage: G.add_vertex(3) 3 sage: G.add_arc(2, 5) Traceback (most recent call last): ... LookupError: vertex (5) is not a vertex of the graph sage: G.add_arc(1, 3) sage: G.has_arc(1, 3) True sage: G.has_arc(2, 3) False - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(3), extra_vertices=Integer(3)) >>> G.add_vertex(Integer(3)) 3 >>> G.add_arc(Integer(2), Integer(5)) Traceback (most recent call last): ... LookupError: vertex (5) is not a vertex of the graph >>> G.add_arc(Integer(1), Integer(3)) >>> G.has_arc(Integer(1), Integer(3)) True >>> G.has_arc(Integer(2), Integer(3)) False - Adding vertices to a dense graph: - sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(3, extra_vertices=3) sage: G.add_vertex(3) 3 sage: G.add_arc(2,5) Traceback (most recent call last): ... LookupError: vertex (5) is not a vertex of the graph sage: G.add_arc(1, 3) sage: G.has_arc(1, 3) True sage: G.has_arc(2, 3) False - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(3), extra_vertices=Integer(3)) >>> G.add_vertex(Integer(3)) 3 >>> G.add_arc(Integer(2),Integer(5)) Traceback (most recent call last): ... LookupError: vertex (5) is not a vertex of the graph >>> G.add_arc(Integer(1), Integer(3)) >>> G.has_arc(Integer(1), Integer(3)) True >>> G.has_arc(Integer(2), Integer(3)) False - Repeatedly adding a vertex using \(k = -1\) will allocate more memory as required: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(3, extra_vertices=0) sage: G.verts() [0, 1, 2] sage: for i in range(10): ....: _ = G.add_vertex(-1); ... sage: G.verts() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(3), extra_vertices=Integer(0)) >>> G.verts() [0, 1, 2] >>> for i in range(Integer(10)): ... _ = G.add_vertex(-Integer(1)); ... >>> G.verts() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] - sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(3, extra_vertices=0) sage: G.verts() [0, 1, 2] sage: for i in range(12): ....: _ = G.add_vertex(-1); ... sage: G.verts() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(3), extra_vertices=Integer(0)) >>> G.verts() [0, 1, 2] >>> for i in range(Integer(12)): ... _ = G.add_vertex(-Integer(1)); ... >>> G.verts() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] 
 - add_vertices(verts)[source]¶
- Add vertices from the iterable - verts.- INPUT: - verts– an iterable of vertices; value -1 has a special meaning – for each such value an unused vertex name is found, used to create a new vertex and returned.
 - OUTPUT: - List of generated labels if there is any -1 in - verts.- Noneotherwise.- See also - add_vertex()– add a vertex to a graph
 - EXAMPLES: - Adding vertices for sparse graphs: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: S = SparseGraph(nverts=4, extra_vertices=4) sage: S.verts() [0, 1, 2, 3] sage: S.add_vertices([3, -1, 4, 9]) [5] sage: S.verts() [0, 1, 2, 3, 4, 5, 9] sage: S.realloc(20) sage: S.verts() [0, 1, 2, 3, 4, 5, 9] - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> S = SparseGraph(nverts=Integer(4), extra_vertices=Integer(4)) >>> S.verts() [0, 1, 2, 3] >>> S.add_vertices([Integer(3), -Integer(1), Integer(4), Integer(9)]) [5] >>> S.verts() [0, 1, 2, 3, 4, 5, 9] >>> S.realloc(Integer(20)) >>> S.verts() [0, 1, 2, 3, 4, 5, 9] - Adding vertices for dense graphs: - sage: from sage.graphs.base.dense_graph import DenseGraph sage: D = DenseGraph(nverts=4, extra_vertices=4) sage: D.verts() [0, 1, 2, 3] sage: D.add_vertices([3, -1, 4, 9]) [5] sage: D.verts() [0, 1, 2, 3, 4, 5, 9] sage: D.realloc(20) sage: D.verts() [0, 1, 2, 3, 4, 5, 9] - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> D = DenseGraph(nverts=Integer(4), extra_vertices=Integer(4)) >>> D.verts() [0, 1, 2, 3] >>> D.add_vertices([Integer(3), -Integer(1), Integer(4), Integer(9)]) [5] >>> D.verts() [0, 1, 2, 3, 4, 5, 9] >>> D.realloc(Integer(20)) >>> D.verts() [0, 1, 2, 3, 4, 5, 9] 
 - all_arcs(u, v)[source]¶
- Give the labels of all arcs - (u, v). An unlabeled arc is interpreted as having label 0.- EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc_label(1,2,1) sage: G.add_arc_label(1,2,2) sage: G.add_arc_label(1,2,2) sage: G.add_arc_label(1,2,2) sage: G.add_arc_label(1,2,3) sage: G.add_arc_label(1,2,3) sage: G.add_arc_label(1,2,4) sage: G.all_arcs(1,2) [4, 3, 3, 2, 2, 2, 1] - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(1)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(2)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(2)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(2)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(3)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(3)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(4)) >>> G.all_arcs(Integer(1),Integer(2)) [4, 3, 3, 2, 2, 2, 1] 
 - arc_label(u, v)[source]¶
- Retrieves the first label found associated with - (u, v).- INPUT: - u,- v– nonnegative integers; must be in self
 - OUTPUT: one of - positive integer – indicates that there is a label on - (u, v)
- 0– either the arc- (u, v)is unlabeled, or there is no arc at all
 - EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc_label(3,4,7) sage: G.arc_label(3,4) 7 - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc_label(Integer(3),Integer(4),Integer(7)) >>> G.arc_label(Integer(3),Integer(4)) 7 - To this function, an unlabeled arc is indistinguishable from a non-arc: - sage: G.add_arc_label(1,0) sage: G.arc_label(1,0) 0 sage: G.arc_label(1,1) 0 - >>> from sage.all import * >>> G.add_arc_label(Integer(1),Integer(0)) >>> G.arc_label(Integer(1),Integer(0)) 0 >>> G.arc_label(Integer(1),Integer(1)) 0 - This function only returns the first label it finds from - uto- v:- sage: G.add_arc_label(1,2,1) sage: G.add_arc_label(1,2,2) sage: G.arc_label(1,2) 2 - >>> from sage.all import * >>> G.add_arc_label(Integer(1),Integer(2),Integer(1)) >>> G.add_arc_label(Integer(1),Integer(2),Integer(2)) >>> G.arc_label(Integer(1),Integer(2)) 2 
 - check_vertex(n)[source]¶
- Check that \(n\) is a vertex of - self.- This method is different from - has_vertex(). The current method raises an error if \(n\) is not a vertex of this graph. On the other hand,- has_vertex()returns a boolean to signify whether or not \(n\) is a vertex of this graph.- INPUT: - n– nonnegative integer representing a vertex
 - OUTPUT: raise an error if \(n\) is not a vertex of this graph - See also - has_vertex()– determine whether this graph has a specific vertex
 - EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: S = SparseGraph(nverts=10, expected_degree=3, extra_vertices=10) sage: S.check_vertex(4) sage: S.check_vertex(12) Traceback (most recent call last): ... LookupError: vertex (12) is not a vertex of the graph sage: S.check_vertex(24) Traceback (most recent call last): ... LookupError: vertex (24) is not a vertex of the graph sage: S.check_vertex(-19) Traceback (most recent call last): ... LookupError: vertex (-19) is not a vertex of the graph - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> S = SparseGraph(nverts=Integer(10), expected_degree=Integer(3), extra_vertices=Integer(10)) >>> S.check_vertex(Integer(4)) >>> S.check_vertex(Integer(12)) Traceback (most recent call last): ... LookupError: vertex (12) is not a vertex of the graph >>> S.check_vertex(Integer(24)) Traceback (most recent call last): ... LookupError: vertex (24) is not a vertex of the graph >>> S.check_vertex(-Integer(19)) Traceback (most recent call last): ... LookupError: vertex (-19) is not a vertex of the graph - sage: from sage.graphs.base.dense_graph import DenseGraph sage: D = DenseGraph(nverts=10, extra_vertices=10) sage: D.check_vertex(4) sage: D.check_vertex(12) Traceback (most recent call last): ... LookupError: vertex (12) is not a vertex of the graph sage: D.check_vertex(24) Traceback (most recent call last): ... LookupError: vertex (24) is not a vertex of the graph sage: D.check_vertex(-19) Traceback (most recent call last): ... LookupError: vertex (-19) is not a vertex of the graph - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> D = DenseGraph(nverts=Integer(10), extra_vertices=Integer(10)) >>> D.check_vertex(Integer(4)) >>> D.check_vertex(Integer(12)) Traceback (most recent call last): ... LookupError: vertex (12) is not a vertex of the graph >>> D.check_vertex(Integer(24)) Traceback (most recent call last): ... LookupError: vertex (24) is not a vertex of the graph >>> D.check_vertex(-Integer(19)) Traceback (most recent call last): ... LookupError: vertex (-19) is not a vertex of the graph 
 - current_allocation()[source]¶
- Report the number of vertices allocated. - OUTPUT: - The number of vertices allocated. This number is usually different from the order of a graph. We may have allocated enough memory for a graph to hold \(n > 0\) vertices, but the order (actual number of vertices) of the graph could be less than \(n\). 
 - EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: S = SparseGraph(nverts=4, extra_vertices=4) sage: S.current_allocation() 8 sage: S.add_vertex(6) 6 sage: S.current_allocation() 8 sage: S.add_vertex(10) 10 sage: S.current_allocation() 16 sage: S.add_vertex(40) Traceback (most recent call last): ... RuntimeError: requested vertex is past twice the allocated range: use realloc sage: S.realloc(50) sage: S.add_vertex(40) 40 sage: S.current_allocation() 50 sage: S.realloc(30) -1 sage: S.current_allocation() 50 sage: S.del_vertex(40) sage: S.realloc(30) sage: S.current_allocation() 30 - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> S = SparseGraph(nverts=Integer(4), extra_vertices=Integer(4)) >>> S.current_allocation() 8 >>> S.add_vertex(Integer(6)) 6 >>> S.current_allocation() 8 >>> S.add_vertex(Integer(10)) 10 >>> S.current_allocation() 16 >>> S.add_vertex(Integer(40)) Traceback (most recent call last): ... RuntimeError: requested vertex is past twice the allocated range: use realloc >>> S.realloc(Integer(50)) >>> S.add_vertex(Integer(40)) 40 >>> S.current_allocation() 50 >>> S.realloc(Integer(30)) -1 >>> S.current_allocation() 50 >>> S.del_vertex(Integer(40)) >>> S.realloc(Integer(30)) >>> S.current_allocation() 30 - The actual number of vertices in a graph might be less than the number of vertices allocated for the graph: - sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(nverts=3, extra_vertices=2) sage: order = len(G.verts()) sage: order 3 sage: G.current_allocation() 5 sage: order < G.current_allocation() True - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(nverts=Integer(3), extra_vertices=Integer(2)) >>> order = len(G.verts()) >>> order 3 >>> G.current_allocation() 5 >>> order < G.current_allocation() True 
 - del_all_arcs(u, v)[source]¶
- Delete all arcs from - uto- v.- INPUT: - u– integer; the tail of an arc
- v– integer; the head of an arc
 - EXAMPLES: - On the - CGraphlevel, this always produces an error, as there are no vertices:- sage: from sage.graphs.base.c_graph import CGraph sage: G = CGraph() sage: G.del_all_arcs(0,1) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraph >>> G = CGraph() >>> G.del_all_arcs(Integer(0),Integer(1)) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - It works, once there are vertices and - del_arc_unsafe()is implemented:- sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc_label(0,1,0) sage: G.add_arc_label(0,1,1) sage: G.add_arc_label(0,1,2) sage: G.add_arc_label(0,1,3) sage: G.del_all_arcs(0,1) sage: G.has_arc(0,1) False sage: G.arc_label(0,1) 0 sage: G.del_all_arcs(0,1) sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(5) sage: G.add_arc(0, 1) sage: G.has_arc(0, 1) True sage: G.del_all_arcs(0, 1) sage: G.has_arc(0, 1) False - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(0)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(1)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(2)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(3)) >>> G.del_all_arcs(Integer(0),Integer(1)) >>> G.has_arc(Integer(0),Integer(1)) False >>> G.arc_label(Integer(0),Integer(1)) 0 >>> G.del_all_arcs(Integer(0),Integer(1)) >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(5)) >>> G.add_arc(Integer(0), Integer(1)) >>> G.has_arc(Integer(0), Integer(1)) True >>> G.del_all_arcs(Integer(0), Integer(1)) >>> G.has_arc(Integer(0), Integer(1)) False 
 - del_arc_label(u, v, l)[source]¶
- Delete an arc - (u, v)with label- l.- INPUT: - u,- v– nonnegative integers; must be in self
- l– positive integer label, or zero for no label
 - EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc_label(0,1,0) sage: G.add_arc_label(0,1,1) sage: G.add_arc_label(0,1,2) sage: G.add_arc_label(0,1,2) sage: G.add_arc_label(0,1,3) sage: G.del_arc_label(0,1,2) sage: G.all_arcs(0,1) [0, 3, 2, 1] sage: G.del_arc_label(0,1,0) sage: G.all_arcs(0,1) [3, 2, 1] - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(0)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(1)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(2)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(2)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(3)) >>> G.del_arc_label(Integer(0),Integer(1),Integer(2)) >>> G.all_arcs(Integer(0),Integer(1)) [0, 3, 2, 1] >>> G.del_arc_label(Integer(0),Integer(1),Integer(0)) >>> G.all_arcs(Integer(0),Integer(1)) [3, 2, 1] 
 - del_vertex(v)[source]¶
- Delete the vertex \(v\), along with all edges incident to it. - If \(v\) is not in - self, fails silently.- INPUT: - v– nonnegative integer representing a vertex
 - See also - del_vertex_unsafe– delete a vertex from a graph. This method is potentially unsafe. Use- del_vertex()instead
 - EXAMPLES: - Deleting vertices of sparse graphs: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(3) sage: G.add_arc(0, 1) sage: G.add_arc(0, 2) sage: G.add_arc(1, 2) sage: G.add_arc(2, 0) sage: G.del_vertex(2) sage: for i in range(2): ....: for j in range(2): ....: if G.has_arc(i, j): ....: print("{} {}".format(i,j)) 0 1 sage: G = SparseGraph(3) sage: G.add_arc(0, 1) sage: G.add_arc(0, 2) sage: G.add_arc(1, 2) sage: G.add_arc(2, 0) sage: G.del_vertex(1) sage: for i in range(3): ....: for j in range(3): ....: if G.has_arc(i, j): ....: print("{} {}".format(i,j)) 0 2 2 0 - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(3)) >>> G.add_arc(Integer(0), Integer(1)) >>> G.add_arc(Integer(0), Integer(2)) >>> G.add_arc(Integer(1), Integer(2)) >>> G.add_arc(Integer(2), Integer(0)) >>> G.del_vertex(Integer(2)) >>> for i in range(Integer(2)): ... for j in range(Integer(2)): ... if G.has_arc(i, j): ... print("{} {}".format(i,j)) 0 1 >>> G = SparseGraph(Integer(3)) >>> G.add_arc(Integer(0), Integer(1)) >>> G.add_arc(Integer(0), Integer(2)) >>> G.add_arc(Integer(1), Integer(2)) >>> G.add_arc(Integer(2), Integer(0)) >>> G.del_vertex(Integer(1)) >>> for i in range(Integer(3)): ... for j in range(Integer(3)): ... if G.has_arc(i, j): ... print("{} {}".format(i,j)) 0 2 2 0 - Deleting vertices of dense graphs: - sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(4) sage: G.add_arc(0, 1); G.add_arc(0, 2) sage: G.add_arc(3, 1); G.add_arc(3, 2) sage: G.add_arc(1, 2) sage: G.verts() [0, 1, 2, 3] sage: G.del_vertex(3); G.verts() [0, 1, 2] sage: for i in range(3): ....: for j in range(3): ....: if G.has_arc(i, j): ....: print("{} {}".format(i,j)) 0 1 0 2 1 2 - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(4)) >>> G.add_arc(Integer(0), Integer(1)); G.add_arc(Integer(0), Integer(2)) >>> G.add_arc(Integer(3), Integer(1)); G.add_arc(Integer(3), Integer(2)) >>> G.add_arc(Integer(1), Integer(2)) >>> G.verts() [0, 1, 2, 3] >>> G.del_vertex(Integer(3)); G.verts() [0, 1, 2] >>> for i in range(Integer(3)): ... for j in range(Integer(3)): ... if G.has_arc(i, j): ... print("{} {}".format(i,j)) 0 1 0 2 1 2 - If the vertex to be deleted is not in this graph, then fail silently: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(3) sage: G.verts() [0, 1, 2] sage: G.has_vertex(3) False sage: G.del_vertex(3) sage: G.verts() [0, 1, 2] - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(3)) >>> G.verts() [0, 1, 2] >>> G.has_vertex(Integer(3)) False >>> G.del_vertex(Integer(3)) >>> G.verts() [0, 1, 2] - sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(5) sage: G.verts() [0, 1, 2, 3, 4] sage: G.has_vertex(6) False sage: G.del_vertex(6) sage: G.verts() [0, 1, 2, 3, 4] - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(5)) >>> G.verts() [0, 1, 2, 3, 4] >>> G.has_vertex(Integer(6)) False >>> G.del_vertex(Integer(6)) >>> G.verts() [0, 1, 2, 3, 4] 
 - has_arc(u, v)[source]¶
- Check if the arc - (u, v)is in this graph.- INPUT: - u– integer; the tail of an arc
- v– integer; the head of an arc
 - OUTPUT: - Print a - Not Implemented!message. This method is not implemented at the- CGraphlevel. A child class should provide a suitable implementation.
 - EXAMPLES: - On the - CGraphthis always returns- False:- sage: from sage.graphs.base.c_graph import CGraph sage: G = CGraph() sage: G.has_arc(0, 1) False - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraph >>> G = CGraph() >>> G.has_arc(Integer(0), Integer(1)) False - It works once - has_arc_unsafeis implemented:- sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(5) sage: G.add_arc(0, 1) sage: G.has_arc(1, 0) False sage: G.has_arc(0, 1) True sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc_label(0,1) sage: G.has_arc(1,0) False sage: G.has_arc(0,1) True - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(5)) >>> G.add_arc(Integer(0), Integer(1)) >>> G.has_arc(Integer(1), Integer(0)) False >>> G.has_arc(Integer(0), Integer(1)) True >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc_label(Integer(0),Integer(1)) >>> G.has_arc(Integer(1),Integer(0)) False >>> G.has_arc(Integer(0),Integer(1)) True 
 - has_arc_label(u, v, l)[source]¶
- Indicates whether there is an arc - (u, v)with label- l.- INPUT: - u,- v– nonnegative integers; must be in self
- l– positive integer label, or zero for no label
 - EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc_label(0,1,0) sage: G.add_arc_label(0,1,1) sage: G.add_arc_label(0,1,2) sage: G.add_arc_label(0,1,2) sage: G.has_arc_label(0,1,1) True sage: G.has_arc_label(0,1,2) True sage: G.has_arc_label(0,1,3) False - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(0)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(1)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(2)) >>> G.add_arc_label(Integer(0),Integer(1),Integer(2)) >>> G.has_arc_label(Integer(0),Integer(1),Integer(1)) True >>> G.has_arc_label(Integer(0),Integer(1),Integer(2)) True >>> G.has_arc_label(Integer(0),Integer(1),Integer(3)) False 
 - has_vertex(n)[source]¶
- Determine whether the vertex \(n\) is in - self.- This method is different from - check_vertex(). The current method returns a boolean to signify whether or not \(n\) is a vertex of this graph. On the other hand,- check_vertex()raises an error if \(n\) is not a vertex of this graph.- INPUT: - n– nonnegative integer representing a vertex
 - OUTPUT: - Trueif \(n\) is a vertex of this graph;- Falseotherwise- See also - check_vertex()– raise an error if this graph does not contain a specific vertex.
 - EXAMPLES: - Upon initialization, a - SparseGraphor- DenseGraphhas the first- nvertsvertices:- sage: from sage.graphs.base.sparse_graph import SparseGraph sage: S = SparseGraph(nverts=10, expected_degree=3, extra_vertices=10) sage: S.has_vertex(6) True sage: S.has_vertex(12) False sage: S.has_vertex(24) False sage: S.has_vertex(-19) False - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> S = SparseGraph(nverts=Integer(10), expected_degree=Integer(3), extra_vertices=Integer(10)) >>> S.has_vertex(Integer(6)) True >>> S.has_vertex(Integer(12)) False >>> S.has_vertex(Integer(24)) False >>> S.has_vertex(-Integer(19)) False - sage: from sage.graphs.base.dense_graph import DenseGraph sage: D = DenseGraph(nverts=10, extra_vertices=10) sage: D.has_vertex(6) True sage: D.has_vertex(12) False sage: D.has_vertex(24) False sage: D.has_vertex(-19) False - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> D = DenseGraph(nverts=Integer(10), extra_vertices=Integer(10)) >>> D.has_vertex(Integer(6)) True >>> D.has_vertex(Integer(12)) False >>> D.has_vertex(Integer(24)) False >>> D.has_vertex(-Integer(19)) False 
 - in_neighbors(v)[source]¶
- Return the list of in-neighbors of the vertex - v.- INPUT: - v– integer representing a vertex of this graph
 - OUTPUT: - Raise - NotImplementedError. This method is not implemented at the- CGraphlevel. A child class should provide a suitable implementation.
 - Note - Due to the implementation of SparseGraph, this method is much more expensive than out_neighbors_unsafe for SparseGraph’s. - EXAMPLES: - On the - CGraphlevel, this always produces an error, as there are no vertices:- sage: from sage.graphs.base.c_graph import CGraph sage: G = CGraph() sage: G.in_neighbors(0) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraph >>> G = CGraph() >>> G.in_neighbors(Integer(0)) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - It works, once there are vertices and - out_neighbors_unsafe()is implemented:- sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(5) sage: G.add_arc(0, 1) sage: G.add_arc(3, 1) sage: G.add_arc(1, 3) sage: G.in_neighbors(1) [0, 3] sage: G.in_neighbors(3) [1] sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc(0,1) sage: G.add_arc(3,1) sage: G.add_arc(1,3) sage: G.in_neighbors(1) [0, 3] sage: G.in_neighbors(3) [1] - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(5)) >>> G.add_arc(Integer(0), Integer(1)) >>> G.add_arc(Integer(3), Integer(1)) >>> G.add_arc(Integer(1), Integer(3)) >>> G.in_neighbors(Integer(1)) [0, 3] >>> G.in_neighbors(Integer(3)) [1] >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc(Integer(0),Integer(1)) >>> G.add_arc(Integer(3),Integer(1)) >>> G.add_arc(Integer(1),Integer(3)) >>> G.in_neighbors(Integer(1)) [0, 3] >>> G.in_neighbors(Integer(3)) [1] 
 - out_neighbors(u)[source]¶
- Return the list of out-neighbors of the vertex - u.- INPUT: - u– integer representing a vertex of this graph
 - EXAMPLES: - On the - CGraphlevel, this always produces an error, as there are no vertices:- sage: from sage.graphs.base.c_graph import CGraph sage: G = CGraph() sage: G.out_neighbors(0) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraph >>> G = CGraph() >>> G.out_neighbors(Integer(0)) Traceback (most recent call last): ... LookupError: vertex (0) is not a vertex of the graph - It works, once there are vertices and - out_neighbors_unsafe()is implemented:- sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(5) sage: G.add_arc(0, 1) sage: G.add_arc(1, 2) sage: G.add_arc(1, 3) sage: G.out_neighbors(0) [1] sage: G.out_neighbors(1) [2, 3] sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc(0,1) sage: G.add_arc(1,2) sage: G.add_arc(1,3) sage: G.out_neighbors(0) [1] sage: G.out_neighbors(1) [2, 3] - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(5)) >>> G.add_arc(Integer(0), Integer(1)) >>> G.add_arc(Integer(1), Integer(2)) >>> G.add_arc(Integer(1), Integer(3)) >>> G.out_neighbors(Integer(0)) [1] >>> G.out_neighbors(Integer(1)) [2, 3] >>> from sage.graphs.base.sparse_graph import SparseGraph >>> G = SparseGraph(Integer(5)) >>> G.add_arc(Integer(0),Integer(1)) >>> G.add_arc(Integer(1),Integer(2)) >>> G.add_arc(Integer(1),Integer(3)) >>> G.out_neighbors(Integer(0)) [1] >>> G.out_neighbors(Integer(1)) [2, 3] 
 - realloc(total)[source]¶
- Reallocate the number of vertices to use, without actually adding any. - INPUT: - total– integer; the total size to make the array of vertices
 - OUTPUT: - Raise a - NotImplementedError. This method is not implemented in this base class. A child class should provide a suitable implementation.
 - See also - EXAMPLES: - First, note that - realloc()is implemented for- SparseGraphand- DenseGraphdifferently, and is not implemented at the- CGraphlevel:- sage: from sage.graphs.base.c_graph import CGraph sage: G = CGraph() sage: G.realloc(20) Traceback (most recent call last): ... NotImplementedError - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraph >>> G = CGraph() >>> G.realloc(Integer(20)) Traceback (most recent call last): ... NotImplementedError - The - reallocimplementation for sparse graphs:- sage: from sage.graphs.base.sparse_graph import SparseGraph sage: S = SparseGraph(nverts=4, extra_vertices=4) sage: S.current_allocation() 8 sage: S.add_vertex(6) 6 sage: S.current_allocation() 8 sage: S.add_vertex(10) 10 sage: S.current_allocation() 16 sage: S.add_vertex(40) Traceback (most recent call last): ... RuntimeError: requested vertex is past twice the allocated range: use realloc sage: S.realloc(50) sage: S.add_vertex(40) 40 sage: S.current_allocation() 50 sage: S.realloc(30) -1 sage: S.current_allocation() 50 sage: S.del_vertex(40) sage: S.realloc(30) sage: S.current_allocation() 30 - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> S = SparseGraph(nverts=Integer(4), extra_vertices=Integer(4)) >>> S.current_allocation() 8 >>> S.add_vertex(Integer(6)) 6 >>> S.current_allocation() 8 >>> S.add_vertex(Integer(10)) 10 >>> S.current_allocation() 16 >>> S.add_vertex(Integer(40)) Traceback (most recent call last): ... RuntimeError: requested vertex is past twice the allocated range: use realloc >>> S.realloc(Integer(50)) >>> S.add_vertex(Integer(40)) 40 >>> S.current_allocation() 50 >>> S.realloc(Integer(30)) -1 >>> S.current_allocation() 50 >>> S.del_vertex(Integer(40)) >>> S.realloc(Integer(30)) >>> S.current_allocation() 30 - The - reallocimplementation for dense graphs:- sage: from sage.graphs.base.dense_graph import DenseGraph sage: D = DenseGraph(nverts=4, extra_vertices=4) sage: D.current_allocation() 8 sage: D.add_vertex(6) 6 sage: D.current_allocation() 8 sage: D.add_vertex(10) 10 sage: D.current_allocation() 16 sage: D.add_vertex(40) Traceback (most recent call last): ... RuntimeError: requested vertex is past twice the allocated range: use realloc sage: D.realloc(50) sage: D.add_vertex(40) 40 sage: D.current_allocation() 50 sage: D.realloc(30) -1 sage: D.current_allocation() 50 sage: D.del_vertex(40) sage: D.realloc(30) sage: D.current_allocation() 30 - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> D = DenseGraph(nverts=Integer(4), extra_vertices=Integer(4)) >>> D.current_allocation() 8 >>> D.add_vertex(Integer(6)) 6 >>> D.current_allocation() 8 >>> D.add_vertex(Integer(10)) 10 >>> D.current_allocation() 16 >>> D.add_vertex(Integer(40)) Traceback (most recent call last): ... RuntimeError: requested vertex is past twice the allocated range: use realloc >>> D.realloc(Integer(50)) >>> D.add_vertex(Integer(40)) 40 >>> D.current_allocation() 50 >>> D.realloc(Integer(30)) -1 >>> D.current_allocation() 50 >>> D.del_vertex(Integer(40)) >>> D.realloc(Integer(30)) >>> D.current_allocation() 30 
 - verts()[source]¶
- Return a list of the vertices in - self.- EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraph sage: S = SparseGraph(nverts=4, extra_vertices=4) sage: S.verts() [0, 1, 2, 3] sage: S.add_vertices([3,5,7,9]) sage: S.verts() [0, 1, 2, 3, 5, 7, 9] sage: S.realloc(20) sage: S.verts() [0, 1, 2, 3, 5, 7, 9] - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraph >>> S = SparseGraph(nverts=Integer(4), extra_vertices=Integer(4)) >>> S.verts() [0, 1, 2, 3] >>> S.add_vertices([Integer(3),Integer(5),Integer(7),Integer(9)]) >>> S.verts() [0, 1, 2, 3, 5, 7, 9] >>> S.realloc(Integer(20)) >>> S.verts() [0, 1, 2, 3, 5, 7, 9] - sage: from sage.graphs.base.dense_graph import DenseGraph sage: G = DenseGraph(3, extra_vertices=2) sage: G.verts() [0, 1, 2] sage: G.del_vertex(0) sage: G.verts() [1, 2] - >>> from sage.all import * >>> from sage.graphs.base.dense_graph import DenseGraph >>> G = DenseGraph(Integer(3), extra_vertices=Integer(2)) >>> G.verts() [0, 1, 2] >>> G.del_vertex(Integer(0)) >>> G.verts() [1, 2] 
 
- class sage.graphs.base.c_graph.CGraphBackend[source]¶
- Bases: - GenericGraphBackend- Base class for sparse and dense graph backends. - sage: from sage.graphs.base.c_graph import CGraphBackend - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraphBackend - This class is extended by - SparseGraphBackendand- DenseGraphBackend, which are fully functional backends. This class is mainly just for vertex functions, which are the same for both. A- CGraphBackendwill not work on its own:- sage: from sage.graphs.base.c_graph import CGraphBackend sage: CGB = CGraphBackend() sage: CGB.degree(0, True) Traceback (most recent call last): ... NotImplementedError: a derived class must return ``self._cg`` - >>> from sage.all import * >>> from sage.graphs.base.c_graph import CGraphBackend >>> CGB = CGraphBackend() >>> CGB.degree(Integer(0), True) Traceback (most recent call last): ... NotImplementedError: a derived class must return ``self._cg`` - The appropriate way to use these backends is via Sage graphs: - sage: G = Graph(30) sage: G.add_edges([(0,1), (0,3), (4,5), (9, 23)]) sage: G.edges(sort=True, labels=False) [(0, 1), (0, 3), (4, 5), (9, 23)] - >>> from sage.all import * >>> G = Graph(Integer(30)) >>> G.add_edges([(Integer(0),Integer(1)), (Integer(0),Integer(3)), (Integer(4),Integer(5)), (Integer(9), Integer(23))]) >>> G.edges(sort=True, labels=False) [(0, 1), (0, 3), (4, 5), (9, 23)] - This class handles the labels of vertices and edges. For vertices it uses two dictionaries - vertex_labelsand- vertex_ints. They are just opposite of each other:- vertex_intsmakes a translation from label to integers (that are internally used) and- vertex_labelsmake the translation from internally used integers to actual labels. This class tries hard to avoid translation if possible. This will work only if the graph is built on integers from \(0\) to \(n-1\) and the vertices are basically added in increasing order.- See also - SparseGraphBackend– backend for sparse graphs.
- DenseGraphBackend– backend for dense graphs.
 - add_edge(u, v, l, directed)[source]¶
- Add the edge - (u,v)to- self.- INPUT: - u,- v– the vertices of the edge
- l– the edge label
- directed– if- False, also add- (v,u)
 - Note - The input - lis ignored if the backend does not support labels.- EXAMPLES: - sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: D.add_edge(0,1,None,False) sage: list(D.iterator_edges(range(9), True)) [(0, 1, None)] - >>> from sage.all import * >>> D = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> D.add_edge(Integer(0),Integer(1),None,False) >>> list(D.iterator_edges(range(Integer(9)), True)) [(0, 1, None)] - sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9) sage: D.add_edge(0, 1, None, False) sage: list(D.iterator_edges(range(9), True)) [(0, 1, None)] - >>> from sage.all import * >>> D = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(9)) >>> D.add_edge(Integer(0), Integer(1), None, False) >>> list(D.iterator_edges(range(Integer(9)), True)) [(0, 1, None)] 
 - add_edges(edges, directed, remove_loops=False)[source]¶
- Add edges from a list. - INPUT: - edges– the edges to be added; can either be of the form- (u,v)or- (u,v,l)
- directed– if- False, add- (v,u)as well as- (u,v)
- remove_loops– if- True, remove loops
 - EXAMPLES: - sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: D.add_edges([(0,1), (2,3), (4,5), (5,6)], False) sage: list(D.iterator_edges(range(9), True)) [(0, 1, None), (2, 3, None), (4, 5, None), (5, 6, None)] - >>> from sage.all import * >>> D = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> D.add_edges([(Integer(0),Integer(1)), (Integer(2),Integer(3)), (Integer(4),Integer(5)), (Integer(5),Integer(6))], False) >>> list(D.iterator_edges(range(Integer(9)), True)) [(0, 1, None), (2, 3, None), (4, 5, None), (5, 6, None)] 
 - add_vertex(name)[source]¶
- Add a vertex to - self.- INPUT: - name– the vertex to be added (must be hashable). If- None, a new name is created
 - OUTPUT: - If - name = None, the new vertex name is returned.- Noneotherwise.
 - See also - add_vertices()– add a bunch of vertices of this graph
- has_vertex()– returns whether or not this graph has a specific vertex
 - EXAMPLES: - sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9) sage: D.add_vertex(10) sage: D.add_vertex([]) Traceback (most recent call last): ... TypeError: unhashable type: 'list' - >>> from sage.all import * >>> D = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(9)) >>> D.add_vertex(Integer(10)) >>> D.add_vertex([]) Traceback (most recent call last): ... TypeError: unhashable type: 'list' - sage: S = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: S.add_vertex(10) sage: S.add_vertex([]) Traceback (most recent call last): ... TypeError: unhashable type: 'list' - >>> from sage.all import * >>> S = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> S.add_vertex(Integer(10)) >>> S.add_vertex([]) Traceback (most recent call last): ... TypeError: unhashable type: 'list' 
 - add_vertices(vertices)[source]¶
- Add vertices to - self.- INPUT: - vertices– iterator of vertex labels; a new name is created, used and returned in the output list for all- Nonevalues in- vertices
 - OUTPUT: - Generated names of new vertices if there is at least one - Nonevalue present in- vertices.- Noneotherwise.- See also - add_vertex()– add a vertex to this graph
 - EXAMPLES: - sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(1) sage: D.add_vertices([1, 2, 3]) sage: D.add_vertices([None] * 4) [4, 5, 6, 7] - >>> from sage.all import * >>> D = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(1)) >>> D.add_vertices([Integer(1), Integer(2), Integer(3)]) >>> D.add_vertices([None] * Integer(4)) [4, 5, 6, 7] - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(0) sage: G.add_vertices([0, 1]) sage: list(G.iterator_verts(None)) [0, 1] sage: list(G.iterator_edges([0, 1], True)) [] - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0)) >>> G.add_vertices([Integer(0), Integer(1)]) >>> list(G.iterator_verts(None)) [0, 1] >>> list(G.iterator_edges([Integer(0), Integer(1)], True)) [] - sage: import sage.graphs.base.dense_graph sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9) sage: D.add_vertices([10, 11, 12]) - >>> from sage.all import * >>> import sage.graphs.base.dense_graph >>> D = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(9)) >>> D.add_vertices([Integer(10), Integer(11), Integer(12)]) 
 - bidirectional_dijkstra(x, y, weight_function=None, distance_flag=False)[source]¶
- Return the shortest path or distance from - xto- yusing a bidirectional version of Dijkstra’s algorithm.- INPUT: - x– the starting vertex in the shortest path from- xto- y
- y– the end vertex in the shortest path from- xto- y
- weight_function– function (default:- None); a function that inputs an edge- (u, v, l)and outputs its weight. If- None, we use the edge label- las a weight, if- lis not- None, else- 1as a weight.
- distance_flag– boolean (default:- False); when set to- True, the shortest path distance from- xto- yis returned instead of the path.
 - OUTPUT: - A list of vertices in the shortest path from - xto- yor distance from- xto- yis returned depending upon the value of parameter- distance_flag
 - EXAMPLES: - sage: G = Graph(graphs.PetersenGraph()) sage: for (u, v) in G.edges(sort=True, labels=None): ....: G.set_edge_label(u, v, 1) sage: G.shortest_path(0, 1, by_weight=True) [0, 1] sage: G.shortest_path_length(0, 1, by_weight=True) 1 sage: G = DiGraph([(1, 2, {'weight':1}), (1, 3, {'weight':5}), (2, 3, {'weight':1})]) sage: G.shortest_path(1, 3, weight_function=lambda e:e[2]['weight']) [1, 2, 3] sage: G.shortest_path_length(1, 3, weight_function=lambda e:e[2]['weight']) 2 - >>> from sage.all import * >>> G = Graph(graphs.PetersenGraph()) >>> for (u, v) in G.edges(sort=True, labels=None): ... G.set_edge_label(u, v, Integer(1)) >>> G.shortest_path(Integer(0), Integer(1), by_weight=True) [0, 1] >>> G.shortest_path_length(Integer(0), Integer(1), by_weight=True) 1 >>> G = DiGraph([(Integer(1), Integer(2), {'weight':Integer(1)}), (Integer(1), Integer(3), {'weight':Integer(5)}), (Integer(2), Integer(3), {'weight':Integer(1)})]) >>> G.shortest_path(Integer(1), Integer(3), weight_function=lambda e:e[Integer(2)]['weight']) [1, 2, 3] >>> G.shortest_path_length(Integer(1), Integer(3), weight_function=lambda e:e[Integer(2)]['weight']) 2 
 - bidirectional_dijkstra_special(x, y, weight_function=None, exclude_vertices=None, exclude_edges=None, include_vertices=None, distance_flag=False, reduced_weight=None)[source]¶
- Return the shortest path or distance from - xto- yusing a bidirectional version of Dijkstra’s algorithm.- This method is an extension of - bidirectional_dijkstra()method enabling to exclude vertices and/or edges from the search for the shortest path between- xand- y.- This method also has - include_verticesoption enabling to include the vertices which will be used to search for the shortest path between- xand- y.- INPUT: - x– the starting vertex in the shortest path from- xto- y
- y– the end vertex in the shortest path from- xto- y
- exclude_vertices– iterable container (default:- None); iterable of vertices to exclude from the graph while calculating the shortest path from- xto- y
- exclude_edges– iterable container (default:- None); iterable of edges to exclude from the graph while calculating the shortest path from- xto- y
- include_vertices– iterable container (default:- None); iterable of vertices to consider in the graph while calculating the shortest path from- xto- y
- weight_function– function (default:- None); a function that inputs an edge- (u, v, l)and outputs its weight. If- None, we use the edge label- las a weight, if- lis not- None, else- 1as a weight.
- distance_flag– boolean (default:- False); when set to- True, the shortest path distance from- xto- yis returned instead of the path.
- reduced_weight– dictionary (default:- None); a dictionary that takes as input an edge- (u, v)and outputs its reduced weight
 - OUTPUT: - A list of vertices in the shortest path from - xto- yor distance from- xto- yis returned depending upon the value of parameter- distance_flag
 - EXAMPLES: - sage: G = Graph([(1, 2, 20), (2, 3, 10), (3, 4, 30), (1, 5, 20), (5, 6, 10), (6, 4, 50), (4, 7, 5)]) sage: G._backend.bidirectional_dijkstra_special(1, 4, weight_function=lambda e:e[2]) [1, 2, 3, 4] sage: G._backend.bidirectional_dijkstra_special(1, 4, weight_function=lambda e:e[2], exclude_vertices=[2], exclude_edges=[(3, 4)]) [1, 5, 6, 4] sage: G._backend.bidirectional_dijkstra_special(1, 4, weight_function=lambda e:e[2], exclude_vertices=[2, 7]) [1, 5, 6, 4] sage: G._backend.bidirectional_dijkstra_special(1, 4, weight_function=lambda e:e[2], exclude_edges=[(5, 6)]) [1, 2, 3, 4] sage: G._backend.bidirectional_dijkstra_special(1, 4, weight_function=lambda e:e[2], include_vertices=[1, 5, 6, 4]) [1, 5, 6, 4] - >>> from sage.all import * >>> G = Graph([(Integer(1), Integer(2), Integer(20)), (Integer(2), Integer(3), Integer(10)), (Integer(3), Integer(4), Integer(30)), (Integer(1), Integer(5), Integer(20)), (Integer(5), Integer(6), Integer(10)), (Integer(6), Integer(4), Integer(50)), (Integer(4), Integer(7), Integer(5))]) >>> G._backend.bidirectional_dijkstra_special(Integer(1), Integer(4), weight_function=lambda e:e[Integer(2)]) [1, 2, 3, 4] >>> G._backend.bidirectional_dijkstra_special(Integer(1), Integer(4), weight_function=lambda e:e[Integer(2)], exclude_vertices=[Integer(2)], exclude_edges=[(Integer(3), Integer(4))]) [1, 5, 6, 4] >>> G._backend.bidirectional_dijkstra_special(Integer(1), Integer(4), weight_function=lambda e:e[Integer(2)], exclude_vertices=[Integer(2), Integer(7)]) [1, 5, 6, 4] >>> G._backend.bidirectional_dijkstra_special(Integer(1), Integer(4), weight_function=lambda e:e[Integer(2)], exclude_edges=[(Integer(5), Integer(6))]) [1, 2, 3, 4] >>> G._backend.bidirectional_dijkstra_special(Integer(1), Integer(4), weight_function=lambda e:e[Integer(2)], include_vertices=[Integer(1), Integer(5), Integer(6), Integer(4)]) [1, 5, 6, 4] 
 - breadth_first_search(v, reverse=False, ignore_direction=False, report_distance=False, edges=False, forbidden_vertices=None)[source]¶
- Return a breadth-first search from vertex - v.- INPUT: - v– a vertex from which to start the breadth-first search
- reverse– boolean (default:- False); this is only relevant to digraphs. If this is a digraph, consider the reversed graph in which the out-neighbors become the in-neighbors and vice versa.
- ignore_direction– boolean (default:- False); this is only relevant to digraphs. If this is a digraph, ignore all orientations and consider the graph as undirected.
- report_distance– boolean (default:- False); if- True, reports pairs- (vertex, distance)where- distanceis the distance from the- startnodes. If- Falseonly the vertices are reported.
- edges– boolean (default:- False); whether to return the edges of the BFS tree in the order of visit or the vertices (default). Edges are directed in root to leaf orientation of the tree.- Note that parameters - edgesand- report_distancecannot be- Truesimultaneously.
- forbidden_vertices– list (default:- None); set of vertices to avoid during the search. The start vertex- vcannot be in this set.
 - ALGORITHM: - Below is a general template for breadth-first search. - Input: A directed or undirected graph \(G = (V, E)\) of order \(n > 0\). A vertex \(s\) from which to start the search. The vertices are numbered from 1 to \(n = |V|\), i.e. \(V = \{1, 2, \dots, n\}\). 
- Output: A list \(D\) of distances of all vertices from \(s\). A tree \(T\) rooted at \(s\). 
 - \(Q \leftarrow [s]\) # a queue of nodes to visit 
- \(D \leftarrow [\infty, \infty, \dots, \infty]\) # \(n\) copies of \(\infty\) 
- \(D[s] \leftarrow 0\) 
- \(T \leftarrow [\,]\) 
- while \(\text{length}(Q) > 0\) do - \(v \leftarrow \text{dequeue}(Q)\) 
- for each \(w \in \text{adj}(v)\) do # for digraphs, use out-neighbor set \(\text{oadj}(v)\) - if \(D[w] = \infty\) then - \(D[w] \leftarrow D[v] + 1\) 
- \(\text{enqueue}(Q, w)\) 
- \(\text{append}(T, vw)\) 
 
 
 
- return \((D, T)\) 
 - See also - breadth_first_search– breadth-first search for generic graphs.
- depth_first_search– depth-first search for generic graphs.
- depth_first_search()– depth-first search for fast compiled graphs.
 - EXAMPLES: - Breadth-first search of the Petersen graph starting at vertex 0: - sage: G = Graph(graphs.PetersenGraph()) sage: list(G.breadth_first_search(0)) [0, 1, 4, 5, 2, 6, 3, 9, 7, 8] - >>> from sage.all import * >>> G = Graph(graphs.PetersenGraph()) >>> list(G.breadth_first_search(Integer(0))) [0, 1, 4, 5, 2, 6, 3, 9, 7, 8] - Visiting European countries using breadth-first search: - sage: G = graphs.EuropeMap(continental=True) sage: list(G.breadth_first_search("Portugal")) ['Portugal', 'Spain', ..., 'Greece'] - >>> from sage.all import * >>> G = graphs.EuropeMap(continental=True) >>> list(G.breadth_first_search("Portugal")) ['Portugal', 'Spain', ..., 'Greece'] - Avoiding some countries: - sage: list(G.breadth_first_search(“Portugal”, ….: forbidden_vertices=[“Germany”,”Italy”])) [‘Portugal’, ‘Spain’, …, ‘Sweden’] 
 - c_graph()[source]¶
- Return the - ._cgand- ._cg_revattributes.- Note - The - ._cg_revattribute has been removed and hence- Noneis returned.- EXAMPLES: - sage: cg,cg_rev = graphs.PetersenGraph()._backend.c_graph() sage: cg <sage.graphs.base.sparse_graph.SparseGraph object at ...> - >>> from sage.all import * >>> cg,cg_rev = graphs.PetersenGraph()._backend.c_graph() >>> cg <sage.graphs.base.sparse_graph.SparseGraph object at ...> 
 - degree(v, directed)[source]¶
- Return the degree of the vertex - v.- INPUT: - v– a vertex of the graph
- directed– boolean; whether to take into account the orientation of this graph in counting the degree of- v
 - OUTPUT: the degree of vertex - v- EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraphBackend sage: B = SparseGraphBackend(7) sage: B.degree(3, False) 0 - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraphBackend >>> B = SparseGraphBackend(Integer(7)) >>> B.degree(Integer(3), False) 0 
 - del_edge(u, v, l, directed)[source]¶
- Delete edge - (u, v, l).- INPUT: - u,- v– the vertices of the edge
- l– the edge label
- directed– if- False, also delete- (v, u, l)
 - Note - The input - lis ignored if the backend does not support labels.- EXAMPLES: - sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: D.add_edges([(0,1), (2,3), (4,5), (5,6)], False) sage: list(D.iterator_edges(range(9), True)) [(0, 1, None), (2, 3, None), (4, 5, None), (5, 6, None)] sage: D.del_edge(0,1,None,True) sage: list(D.iterator_out_edges(range(9), True)) [(1, 0, None), (2, 3, None), (3, 2, None), (4, 5, None), (5, 4, None), (5, 6, None), (6, 5, None)] - >>> from sage.all import * >>> D = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> D.add_edges([(Integer(0),Integer(1)), (Integer(2),Integer(3)), (Integer(4),Integer(5)), (Integer(5),Integer(6))], False) >>> list(D.iterator_edges(range(Integer(9)), True)) [(0, 1, None), (2, 3, None), (4, 5, None), (5, 6, None)] >>> D.del_edge(Integer(0),Integer(1),None,True) >>> list(D.iterator_out_edges(range(Integer(9)), True)) [(1, 0, None), (2, 3, None), (3, 2, None), (4, 5, None), (5, 4, None), (5, 6, None), (6, 5, None)] - sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9) sage: D.add_edges([(0, 1), (2, 3), (4, 5), (5, 6)], False) sage: list(D.iterator_edges(range(9), True)) [(0, 1, None), (2, 3, None), (4, 5, None), (5, 6, None)] sage: D.del_edge(0, 1, None, True) sage: list(D.iterator_out_edges(range(9), True)) [(1, 0, None), (2, 3, None), (3, 2, None), (4, 5, None), (5, 4, None), (5, 6, None), (6, 5, None)] - >>> from sage.all import * >>> D = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(9)) >>> D.add_edges([(Integer(0), Integer(1)), (Integer(2), Integer(3)), (Integer(4), Integer(5)), (Integer(5), Integer(6))], False) >>> list(D.iterator_edges(range(Integer(9)), True)) [(0, 1, None), (2, 3, None), (4, 5, None), (5, 6, None)] >>> D.del_edge(Integer(0), Integer(1), None, True) >>> list(D.iterator_out_edges(range(Integer(9)), True)) [(1, 0, None), (2, 3, None), (3, 2, None), (4, 5, None), (5, 4, None), (5, 6, None), (6, 5, None)] 
 - del_edges(edges, directed)[source]¶
- Delete edges from a list. - INPUT: - edges– the edges to be added; can either be of the form- (u,v)or- (u,v,l)
- directed– if- False, remove``(v,u)`` as well as- (u,v)
 - EXAMPLES: - sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: D.add_edges([(0,1), (2,3), (4,5), (5,6)], False) sage: D.del_edges([(0,1), (2,3), (4,5), (5,6)], False) sage: list(D.iterator_edges(range(9), True)) [] - >>> from sage.all import * >>> D = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> D.add_edges([(Integer(0),Integer(1)), (Integer(2),Integer(3)), (Integer(4),Integer(5)), (Integer(5),Integer(6))], False) >>> D.del_edges([(Integer(0),Integer(1)), (Integer(2),Integer(3)), (Integer(4),Integer(5)), (Integer(5),Integer(6))], False) >>> list(D.iterator_edges(range(Integer(9)), True)) [] 
 - del_vertex(v)[source]¶
- Delete a vertex in - self, failing silently if the vertex is not in the graph.- INPUT: - v– vertex to be deleted
 - See also - del_vertices()– delete a bunch of vertices from this graph
 - EXAMPLES: - sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9) sage: D.del_vertex(0) sage: D.has_vertex(0) False - >>> from sage.all import * >>> D = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(9)) >>> D.del_vertex(Integer(0)) >>> D.has_vertex(Integer(0)) False - sage: S = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: S.del_vertex(0) sage: S.has_vertex(0) False - >>> from sage.all import * >>> S = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> S.del_vertex(Integer(0)) >>> S.has_vertex(Integer(0)) False 
 - del_vertices(vertices)[source]¶
- Delete vertices from an iterable container. - INPUT: - vertices– iterator of vertex labels
 - OUTPUT: same as for - del_vertex()- See also - del_vertex()– delete a vertex of this graph
 - EXAMPLES: - sage: import sage.graphs.base.dense_graph sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9) sage: D.del_vertices([7, 8]) sage: D.has_vertex(7) False sage: D.has_vertex(6) True - >>> from sage.all import * >>> import sage.graphs.base.dense_graph >>> D = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(9)) >>> D.del_vertices([Integer(7), Integer(8)]) >>> D.has_vertex(Integer(7)) False >>> D.has_vertex(Integer(6)) True - sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: D.del_vertices([1, 2, 3]) sage: D.has_vertex(1) False sage: D.has_vertex(0) True - >>> from sage.all import * >>> D = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> D.del_vertices([Integer(1), Integer(2), Integer(3)]) >>> D.has_vertex(Integer(1)) False >>> D.has_vertex(Integer(0)) True 
 - depth_first_search(v, reverse=False, ignore_direction=False, forbidden_vertices=None)[source]¶
- Return a depth-first search from vertex - v.- INPUT: - v– a vertex from which to start the depth-first search
- reverse– boolean (default:- False); this is only relevant to digraphs. If this is a digraph, consider the reversed graph in which the out-neighbors become the in-neighbors and vice versa.
- ignore_direction– boolean (default:- False); this is only relevant to digraphs. If this is a digraph, ignore all orientations and consider the graph as undirected.
- forbidden_vertices– list (default:- None); set of vertices to avoid during the search. The start vertex- vcannot be in this set.
 - ALGORITHM: - Below is a general template for depth-first search. - Input: A directed or undirected graph \(G = (V, E)\) of order \(n > 0\). A vertex \(s\) from which to start the search. The vertices are numbered from 1 to \(n = |V|\), i.e. \(V = \{1, 2, \dots, n\}\). 
- Output: A list \(D\) of distances of all vertices from \(s\). A tree \(T\) rooted at \(s\). 
 - \(S \leftarrow [s]\) # a stack of nodes to visit 
- \(D \leftarrow [\infty, \infty, \dots, \infty]\) # \(n\) copies of \(\infty\) 
- \(D[s] \leftarrow 0\) 
- \(T \leftarrow [\,]\) 
- while \(\text{length}(S) > 0\) do - \(v \leftarrow \text{pop}(S)\) 
- for each \(w \in \text{adj}(v)\) do # for digraphs, use out-neighbor set \(\text{oadj}(v)\) - if \(D[w] = \infty\) then - \(D[w] \leftarrow D[v] + 1\) 
- \(\text{push}(S, w)\) 
- \(\text{append}(T, vw)\) 
 
 
 
- return \((D, T)\) 
 - See also - breadth_first_search()– breadth-first search for fast compiled graphs.
- breadth_first_search– breadth-first search for generic graphs.
- depth_first_search– depth-first search for generic graphs.
 - EXAMPLES: - Traversing the Petersen graph using depth-first search: - sage: G = graphs.PetersenGraph() sage: list(G.depth_first_search(0)) [0, 5, 8, 6, 9, 7, 2, 3, 4, 1] - >>> from sage.all import * >>> G = graphs.PetersenGraph() >>> list(G.depth_first_search(Integer(0))) [0, 5, 8, 6, 9, 7, 2, 3, 4, 1] - Visiting German cities using depth-first search: - sage: G = Graph({"Mannheim": ["Frankfurt","Karlsruhe"], ....: "Frankfurt": ["Mannheim","Wurzburg","Kassel"], ....: "Kassel": ["Frankfurt","Munchen"], ....: "Munchen": ["Kassel","Nurnberg","Augsburg"], ....: "Augsburg": ["Munchen","Karlsruhe"], ....: "Karlsruhe": ["Mannheim","Augsburg"], ....: "Wurzburg": ["Frankfurt","Erfurt","Nurnberg"], ....: "Nurnberg": ["Wurzburg","Stuttgart","Munchen"], ....: "Stuttgart": ["Nurnberg"], "Erfurt": ["Wurzburg"]}) sage: list(G.depth_first_search("Stuttgart")) ['Stuttgart', 'Nurnberg', ...] - >>> from sage.all import * >>> G = Graph({"Mannheim": ["Frankfurt","Karlsruhe"], ... "Frankfurt": ["Mannheim","Wurzburg","Kassel"], ... "Kassel": ["Frankfurt","Munchen"], ... "Munchen": ["Kassel","Nurnberg","Augsburg"], ... "Augsburg": ["Munchen","Karlsruhe"], ... "Karlsruhe": ["Mannheim","Augsburg"], ... "Wurzburg": ["Frankfurt","Erfurt","Nurnberg"], ... "Nurnberg": ["Wurzburg","Stuttgart","Munchen"], ... "Stuttgart": ["Nurnberg"], "Erfurt": ["Wurzburg"]}) >>> list(G.depth_first_search("Stuttgart")) ['Stuttgart', 'Nurnberg', ...] - Avoiding some cities: - sage: list(G.depth_first_search(“Stuttgart”, ….: forbidden_vertices=[“Frankfurt”, “Munchen”])) [‘Stuttgart’, ‘Nurnberg’, ‘Wurzburg’, ‘Erfurt’] 
 - has_vertex(v)[source]¶
- Check whether - vis a vertex of- self.- INPUT: - v– any object
 - OUTPUT: - Trueif- vis a vertex of this graph;- Falseotherwise- EXAMPLES: - sage: from sage.graphs.base.sparse_graph import SparseGraphBackend sage: B = SparseGraphBackend(7) sage: B.has_vertex(6) True sage: B.has_vertex(7) False - >>> from sage.all import * >>> from sage.graphs.base.sparse_graph import SparseGraphBackend >>> B = SparseGraphBackend(Integer(7)) >>> B.has_vertex(Integer(6)) True >>> B.has_vertex(Integer(7)) False 
 - in_degree(v)[source]¶
- Return the in-degree of - v.- INPUT: - v– a vertex of the graph
 - EXAMPLES: - sage: D = DiGraph( { 0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1] } ) sage: D.out_degree(1) 2 - >>> from sage.all import * >>> D = DiGraph( { Integer(0): [Integer(1),Integer(2),Integer(3)], Integer(1): [Integer(0),Integer(2)], Integer(2): [Integer(3)], Integer(3): [Integer(4)], Integer(4): [Integer(0),Integer(5)], Integer(5): [Integer(1)] } ) >>> D.out_degree(Integer(1)) 2 
 - is_connected(forbidden_vertices=None)[source]¶
- Check whether the graph is connected. - INPUT: - forbidden_vertices– list (default:- None); set of vertices to avoid during the search
 - EXAMPLES: - Petersen’s graph is connected: - sage: DiGraph(graphs.PetersenGraph()).is_connected() True - >>> from sage.all import * >>> DiGraph(graphs.PetersenGraph()).is_connected() True - While the disjoint union of two of them is not: - sage: DiGraph(2*graphs.PetersenGraph()).is_connected() False - >>> from sage.all import * >>> DiGraph(Integer(2)*graphs.PetersenGraph()).is_connected() False - A graph with non-integer vertex labels: - sage: Graph(graphs.CubeGraph(3)).is_connected() True - >>> from sage.all import * >>> Graph(graphs.CubeGraph(Integer(3))).is_connected() True - A graph with forbidden vertices: - sage: G = graphs.PathGraph(5) sage: G._backend.is_connected() True sage: G._backend.is_connected(forbidden_vertices=[1]) False sage: G._backend.is_connected(forbidden_vertices=[0, 1]) True - >>> from sage.all import * >>> G = graphs.PathGraph(Integer(5)) >>> G._backend.is_connected() True >>> G._backend.is_connected(forbidden_vertices=[Integer(1)]) False >>> G._backend.is_connected(forbidden_vertices=[Integer(0), Integer(1)]) True 
 - is_directed_acyclic(certificate=False)[source]¶
- Check whether the graph is both directed and acyclic (possibly with a certificate) - INPUT: - certificate– boolean (default:- False); whether to return a certificate
 - OUTPUT: - When - certificate=False, returns a boolean value. When- certificate=True:- If the graph is acyclic, returns a pair - (True, ordering)where- orderingis a list of the vertices such that- uappears before- vin- orderingif- u, vis an edge.
- Else, returns a pair - (False, cycle)where- cycleis a list of vertices representing a circuit in the graph.
 - ALGORITHM: - We pick a vertex at random, think hard and find out that if we are to remove the vertex from the graph we must remove all of its out-neighbors in the first place. So we put all of its out-neighbours in a stack, and repeat the same procedure with the vertex on top of the stack (when a vertex on top of the stack has no out-neighbors, we remove it immediately). Of course, for each vertex we only add its outneighbors to the end of the stack once : if for some reason the previous algorithm leads us to do it twice, it means we have found a circuit. - We keep track of the vertices whose out-neighborhood has been added to the stack once with a variable named - tried.- There is no reason why the graph should be empty at the end of this procedure, so we run it again on the remaining vertices until none are left or a circuit is found. - Note - The graph is assumed to be directed. An exception is raised if it is not. - EXAMPLES: - At first, the following graph is acyclic: - sage: D = DiGraph({ 0:[1,2,3], 4:[2,5], 1:[8], 2:[7], 3:[7], 5:[6,7], 7:[8], 6:[9], 8:[10], 9:[10] }) sage: D.plot(layout='circular').show() # needs sage.plot sage: D.is_directed_acyclic() True - >>> from sage.all import * >>> D = DiGraph({ Integer(0):[Integer(1),Integer(2),Integer(3)], Integer(4):[Integer(2),Integer(5)], Integer(1):[Integer(8)], Integer(2):[Integer(7)], Integer(3):[Integer(7)], Integer(5):[Integer(6),Integer(7)], Integer(7):[Integer(8)], Integer(6):[Integer(9)], Integer(8):[Integer(10)], Integer(9):[Integer(10)] }) >>> D.plot(layout='circular').show() # needs sage.plot >>> D.is_directed_acyclic() True - Adding an edge from \(9\) to \(7\) does not change it: - sage: D.add_edge(9,7) sage: D.is_directed_acyclic() True - >>> from sage.all import * >>> D.add_edge(Integer(9),Integer(7)) >>> D.is_directed_acyclic() True - We can obtain as a proof an ordering of the vertices such that \(u\) appears before \(v\) if \(uv\) is an edge of the graph: - sage: D.is_directed_acyclic(certificate = True) (True, [4, 5, 6, 9, 0, 1, 2, 3, 7, 8, 10]) - >>> from sage.all import * >>> D.is_directed_acyclic(certificate = True) (True, [4, 5, 6, 9, 0, 1, 2, 3, 7, 8, 10]) - Adding an edge from 7 to 4, though, makes a difference: - sage: D.add_edge(7,4) sage: D.is_directed_acyclic() False - >>> from sage.all import * >>> D.add_edge(Integer(7),Integer(4)) >>> D.is_directed_acyclic() False - Indeed, it creates a circuit \(7, 4, 5\): - sage: D.is_directed_acyclic(certificate = True) (False, [7, 4, 5]) - >>> from sage.all import * >>> D.is_directed_acyclic(certificate = True) (False, [7, 4, 5]) - Checking acyclic graphs are indeed acyclic - sage: def random_acyclic(n, p): ....: g = graphs.RandomGNP(n, p) ....: h = DiGraph() ....: h.add_edges([ ((u,v) if u<v else (v,u)) for u,v,_ in g.edges(sort=True) ]) ....: return h ... sage: all( random_acyclic(100, .2).is_directed_acyclic() # long time ....: for i in range(50)) True - >>> from sage.all import * >>> def random_acyclic(n, p): ... g = graphs.RandomGNP(n, p) ... h = DiGraph() ... h.add_edges([ ((u,v) if u<v else (v,u)) for u,v,_ in g.edges(sort=True) ]) ... return h ... >>> all( random_acyclic(Integer(100), RealNumber('.2')).is_directed_acyclic() # long time ... for i in range(Integer(50))) True 
 - is_strongly_connected()[source]¶
- Check whether the graph is strongly connected. - EXAMPLES: - The circuit on 3 vertices is obviously strongly connected: - sage: g = DiGraph({0: [1], 1: [2], 2: [0]}) sage: g.is_strongly_connected() True - >>> from sage.all import * >>> g = DiGraph({Integer(0): [Integer(1)], Integer(1): [Integer(2)], Integer(2): [Integer(0)]}) >>> g.is_strongly_connected() True - But a transitive triangle is not: - sage: g = DiGraph({0: [1,2], 1: [2]}) sage: g.is_strongly_connected() False - >>> from sage.all import * >>> g = DiGraph({Integer(0): [Integer(1),Integer(2)], Integer(1): [Integer(2)]}) >>> g.is_strongly_connected() False 
 - is_subgraph(other, vertices, ignore_labels=False)[source]¶
- Return whether the subgraph of - selfinduced by- verticesis a subgraph of- other.- If - verticesare the vertices of- self, return whether- selfis a subgraph of- other.- INPUT: - other– a subclass of- CGraphBackend
- vertices– a iterable over the vertex labels
- ignore_labels– boolean (default:- False); whether to ignore the labels
 - EXAMPLES: - sage: G = sage.graphs.base.dense_graph.DenseGraphBackend(4, directed=True) sage: H = sage.graphs.base.dense_graph.DenseGraphBackend(4, directed=True) sage: G.add_edges([[0,1],[0,2],[0,3],[1,2]], True) sage: H.add_edges([[0,1],[0,2],[0,3]], True) sage: G.is_subgraph(H, range(4)) False sage: H.is_subgraph(G, range(4)) True sage: G.is_subgraph(H, [0,1,3]) True - >>> from sage.all import * >>> G = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(4), directed=True) >>> H = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(4), directed=True) >>> G.add_edges([[Integer(0),Integer(1)],[Integer(0),Integer(2)],[Integer(0),Integer(3)],[Integer(1),Integer(2)]], True) >>> H.add_edges([[Integer(0),Integer(1)],[Integer(0),Integer(2)],[Integer(0),Integer(3)]], True) >>> G.is_subgraph(H, range(Integer(4))) False >>> H.is_subgraph(G, range(Integer(4))) True >>> G.is_subgraph(H, [Integer(0),Integer(1),Integer(3)]) True - Ignore the labels or not: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(3, directed=True) sage: G.multiple_edges(True) sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(3, directed=True) sage: H.multiple_edges(True) sage: G.add_edges([[0,1,'a'], [0,1,'b'], [0,2,'c'], [0,2,'d'], [0,2,'e']], True) sage: H.add_edges([[0,1,'a'], [0,1,'foo'], [0,2,'c'], [0,2,'d'], [0,2,'e'], [0,2,'e']], True) sage: G.is_subgraph(H, range(3)) False sage: G.is_subgraph(H, range(3), ignore_labels=True) True - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(3), directed=True) >>> G.multiple_edges(True) >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(3), directed=True) >>> H.multiple_edges(True) >>> G.add_edges([[Integer(0),Integer(1),'a'], [Integer(0),Integer(1),'b'], [Integer(0),Integer(2),'c'], [Integer(0),Integer(2),'d'], [Integer(0),Integer(2),'e']], True) >>> H.add_edges([[Integer(0),Integer(1),'a'], [Integer(0),Integer(1),'foo'], [Integer(0),Integer(2),'c'], [Integer(0),Integer(2),'d'], [Integer(0),Integer(2),'e'], [Integer(0),Integer(2),'e']], True) >>> G.is_subgraph(H, range(Integer(3))) False >>> G.is_subgraph(H, range(Integer(3)), ignore_labels=True) True - Multiplicities of edges are considered: - sage: G.is_subgraph(H, [0,2]) True sage: H.is_subgraph(G, [0,2]) False - >>> from sage.all import * >>> G.is_subgraph(H, [Integer(0),Integer(2)]) True >>> H.is_subgraph(G, [Integer(0),Integer(2)]) False 
 - iterator_edges(vertices, labels)[source]¶
- Iterate over the edges incident to a sequence of vertices. - Edges are assumed to be undirected. - Warning - This will try to sort the two ends of every edge. - INPUT: - vertices– list of vertex labels
- labels– boolean, whether to return labels as well
 - EXAMPLES: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: G.add_edge(1,2,3,False) sage: list(G.iterator_edges(range(9), False)) [(1, 2)] sage: list(G.iterator_edges(range(9), True)) [(1, 2, 3)] - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> G.add_edge(Integer(1),Integer(2),Integer(3),False) >>> list(G.iterator_edges(range(Integer(9)), False)) [(1, 2)] >>> list(G.iterator_edges(range(Integer(9)), True)) [(1, 2, 3)] 
 - iterator_in_edges(vertices, labels)[source]¶
- Iterate over the incoming edges incident to a sequence of vertices. - INPUT: - vertices– list of vertex labels
- labels– boolean, whether to return labels as well
 - EXAMPLES: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: G.add_edge(1,2,3,True) sage: list(G.iterator_in_edges([1], False)) [] sage: list(G.iterator_in_edges([2], False)) [(1, 2)] sage: list(G.iterator_in_edges([2], True)) [(1, 2, 3)] - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> G.add_edge(Integer(1),Integer(2),Integer(3),True) >>> list(G.iterator_in_edges([Integer(1)], False)) [] >>> list(G.iterator_in_edges([Integer(2)], False)) [(1, 2)] >>> list(G.iterator_in_edges([Integer(2)], True)) [(1, 2, 3)] 
 - iterator_in_nbrs(v)[source]¶
- Iterate over the incoming neighbors of - v.- INPUT: - v– a vertex of this graph
 - OUTPUT: an iterator over the in-neighbors of the vertex - v- See also - iterator_nbrs()– returns an iterator over the neighbors of a vertex
- iterator_out_nbrs()– returns an iterator over the out-neighbors of a vertex
 - EXAMPLES: - sage: P = DiGraph(graphs.PetersenGraph().to_directed()) sage: list(P._backend.iterator_in_nbrs(0)) [1, 4, 5] - >>> from sage.all import * >>> P = DiGraph(graphs.PetersenGraph().to_directed()) >>> list(P._backend.iterator_in_nbrs(Integer(0))) [1, 4, 5] 
 - iterator_nbrs(v)[source]¶
- Iterate over the neighbors of - v.- INPUT: - v– a vertex of this graph
 - OUTPUT: an iterator over the neighbors the vertex - v- See also - iterator_in_nbrs()– returns an iterator over the in-neighbors of a vertex
- iterator_out_nbrs()– returns an iterator over the out-neighbors of a vertex
- iterator_verts()– returns an iterator over a given set of vertices
 - EXAMPLES: - sage: P = Graph(graphs.PetersenGraph()) sage: list(P._backend.iterator_nbrs(0)) [1, 4, 5] sage: Q = DiGraph(P) sage: list(Q._backend.iterator_nbrs(0)) [1, 4, 5] - >>> from sage.all import * >>> P = Graph(graphs.PetersenGraph()) >>> list(P._backend.iterator_nbrs(Integer(0))) [1, 4, 5] >>> Q = DiGraph(P) >>> list(Q._backend.iterator_nbrs(Integer(0))) [1, 4, 5] 
 - iterator_out_edges(vertices, labels)[source]¶
- Iterate over the outbound edges incident to a sequence of vertices. - INPUT: - vertices– list of vertex labels
- labels– boolean, whether to return labels as well
 - EXAMPLES: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: G.add_edge(1,2,3,True) sage: list(G.iterator_out_edges([2], False)) [] sage: list(G.iterator_out_edges([1], False)) [(1, 2)] sage: list(G.iterator_out_edges([1], True)) [(1, 2, 3)] - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> G.add_edge(Integer(1),Integer(2),Integer(3),True) >>> list(G.iterator_out_edges([Integer(2)], False)) [] >>> list(G.iterator_out_edges([Integer(1)], False)) [(1, 2)] >>> list(G.iterator_out_edges([Integer(1)], True)) [(1, 2, 3)] 
 - iterator_out_nbrs(v)[source]¶
- Iterate over the outgoing neighbors of - v.- INPUT: - v– a vertex of this graph
 - OUTPUT: an iterator over the out-neighbors of the vertex - v- See also - iterator_nbrs()– returns an iterator over the neighbors of a vertex
- iterator_in_nbrs()– returns an iterator over the in-neighbors of a vertex
 - EXAMPLES: - sage: P = DiGraph(graphs.PetersenGraph().to_directed()) sage: list(P._backend.iterator_out_nbrs(0)) [1, 4, 5] - >>> from sage.all import * >>> P = DiGraph(graphs.PetersenGraph().to_directed()) >>> list(P._backend.iterator_out_nbrs(Integer(0))) [1, 4, 5] 
 - iterator_unsorted_edges(vertices, labels)[source]¶
- Iterate over the edges incident to a sequence of vertices. - Edges are assumed to be undirected. - This does not sort the ends of each edge. - INPUT: - vertices– list of vertex labels
- labels– boolean, whether to return labels as well
 - EXAMPLES: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: G.add_edge(1,2,3,False) sage: list(G.iterator_unsorted_edges(range(9), False)) [(2, 1)] sage: list(G.iterator_unsorted_edges(range(9), True)) [(2, 1, 3)] - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(9)) >>> G.add_edge(Integer(1),Integer(2),Integer(3),False) >>> list(G.iterator_unsorted_edges(range(Integer(9)), False)) [(2, 1)] >>> list(G.iterator_unsorted_edges(range(Integer(9)), True)) [(2, 1, 3)] 
 - iterator_verts(verts=None)[source]¶
- Iterate over the vertices of - selfintersected with- verts.- INPUT: - verts– an iterable container of objects (default:- None)
 - OUTPUT: - If - verts=None, return an iterator over all vertices of this graph
- If - vertsis a single vertex of the graph, treat it as the container- [verts]
- If - vertsis a iterable container of vertices, find the intersection of- vertswith the vertex set of this graph and return an iterator over the resulting intersection
 - See also - iterator_nbrs()– returns an iterator over the neighbors of a vertex.
 - EXAMPLES: - sage: P = Graph(graphs.PetersenGraph()) sage: list(P._backend.iterator_verts(P)) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] sage: list(P._backend.iterator_verts()) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] sage: list(P._backend.iterator_verts([1, 2, 3])) [1, 2, 3] sage: list(P._backend.iterator_verts([1, 2, 10])) [1, 2] - >>> from sage.all import * >>> P = Graph(graphs.PetersenGraph()) >>> list(P._backend.iterator_verts(P)) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> list(P._backend.iterator_verts()) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> list(P._backend.iterator_verts([Integer(1), Integer(2), Integer(3)])) [1, 2, 3] >>> list(P._backend.iterator_verts([Integer(1), Integer(2), Integer(10)])) [1, 2] 
 - loops(new=None)[source]¶
- Check whether loops are allowed in this graph. - INPUT: - new– boolean (default:- None); to set or- Noneto get
 - OUTPUT: - If - new=None, return- Trueif this graph allows self-loops or- Falseif self-loops are not allowed
- If - newis a boolean, set the self-loop permission of this graph according to the boolean value of- new
 - EXAMPLES: - sage: G = Graph() sage: G._backend.loops() False sage: G._backend.loops(True) sage: G._backend.loops() True - >>> from sage.all import * >>> G = Graph() >>> G._backend.loops() False >>> G._backend.loops(True) >>> G._backend.loops() True 
 - num_edges(directed)[source]¶
- Return the number of edges in - self.- INPUT: - directed– boolean; whether to count- (u, v)and- (v, u)as one or two edges
 - OUTPUT: - If - directed=True, counts the number of directed edges in this graph. Otherwise, return the size of this graph.
 - See also - num_verts()– return the order of this graph.
 - EXAMPLES: - sage: G = Graph(graphs.PetersenGraph()) sage: G._backend.num_edges(False) 15 - >>> from sage.all import * >>> G = Graph(graphs.PetersenGraph()) >>> G._backend.num_edges(False) 15 
 - num_verts()[source]¶
- Return the number of vertices in - self.- OUTPUT: the order of this graph - See also - num_edges()– return the number of (directed) edges in this graph.
 - EXAMPLES: - sage: G = Graph(graphs.PetersenGraph()) sage: G._backend.num_verts() 10 - >>> from sage.all import * >>> G = Graph(graphs.PetersenGraph()) >>> G._backend.num_verts() 10 
 - out_degree(v)[source]¶
- Return the out-degree of - v.- INPUT: - v– a vertex of the graph
 - EXAMPLES: - sage: D = DiGraph( { 0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1] } ) sage: D.out_degree(1) 2 - >>> from sage.all import * >>> D = DiGraph( { Integer(0): [Integer(1),Integer(2),Integer(3)], Integer(1): [Integer(0),Integer(2)], Integer(2): [Integer(3)], Integer(3): [Integer(4)], Integer(4): [Integer(0),Integer(5)], Integer(5): [Integer(1)] } ) >>> D.out_degree(Integer(1)) 2 
 - relabel(perm, directed)[source]¶
- Relabel the graph according to - perm.- INPUT: - perm– anything which represents a permutation as- v --> perm[v], for example a dict or a list
- directed– ignored (this is here for compatibility with other backends)
 - EXAMPLES: - sage: G = Graph(graphs.PetersenGraph()) sage: G._backend.relabel(range(9,-1,-1), False) sage: G.edges(sort=True) [(0, 2, None), (0, 3, None), (0, 5, None), (1, 3, None), (1, 4, None), (1, 6, None), (2, 4, None), (2, 7, None), (3, 8, None), (4, 9, None), (5, 6, None), (5, 9, None), (6, 7, None), (7, 8, None), (8, 9, None)] - >>> from sage.all import * >>> G = Graph(graphs.PetersenGraph()) >>> G._backend.relabel(range(Integer(9),-Integer(1),-Integer(1)), False) >>> G.edges(sort=True) [(0, 2, None), (0, 3, None), (0, 5, None), (1, 3, None), (1, 4, None), (1, 6, None), (2, 4, None), (2, 7, None), (3, 8, None), (4, 9, None), (5, 6, None), (5, 9, None), (6, 7, None), (7, 8, None), (8, 9, None)] 
 - shortest_path(x, y, distance_flag=False)[source]¶
- Return the shortest path or distance from - xto- y.- INPUT: - x– the starting vertex in the shortest path from- xto- y
- y– the end vertex in the shortest path from- xto- y
- distance_flag– boolean (default:- False); when set to- True, the shortest path distance from- xto- yis returned instead of the path
 - OUTPUT: - A list of vertices in the shortest path from - xto- yor distance from- xto- yis returned depending upon the value of parameter- distance_flag
 - EXAMPLES: - sage: G = Graph(graphs.PetersenGraph()) sage: G.shortest_path(0, 1) [0, 1] sage: G.shortest_path_length(0, 1) 1 - >>> from sage.all import * >>> G = Graph(graphs.PetersenGraph()) >>> G.shortest_path(Integer(0), Integer(1)) [0, 1] >>> G.shortest_path_length(Integer(0), Integer(1)) 1 
 - shortest_path_all_vertices(v, cutoff=None, distance_flag=False)[source]¶
- Return for each reachable vertex - ua shortest- v-upath or distance from- vto- u.- INPUT: - v– a starting vertex in the shortest path
- cutoff– integer (default:- None); maximal distance of returned paths (longer paths will not be returned), ignored when set to- None
- distance_flag– boolean (default:- False); when set to- True, each vertex- uconnected to- vis mapped to shortest path distance from- vto- uinstead of the shortest path in the output dictionary.
 - OUTPUT: - A dictionary which maps each vertex - uconnected to- vto the shortest path list or distance from- vto- udepending upon the value of parameter- distance_flag
 - Note - The weight of edges is not taken into account. - ALGORITHM: - This is just a breadth-first search. - EXAMPLES: - On the Petersen Graph: - sage: g = graphs.PetersenGraph() sage: paths = g._backend.shortest_path_all_vertices(0) sage: all((not paths[v] or len(paths[v])-1 == g.distance(0,v)) for v in g) True sage: g._backend.shortest_path_all_vertices(0, distance_flag=True) {0: 0, 1: 1, 2: 2, 3: 2, 4: 1, 5: 1, 6: 2, 7: 2, 8: 2, 9: 2} - >>> from sage.all import * >>> g = graphs.PetersenGraph() >>> paths = g._backend.shortest_path_all_vertices(Integer(0)) >>> all((not paths[v] or len(paths[v])-Integer(1) == g.distance(Integer(0),v)) for v in g) True >>> g._backend.shortest_path_all_vertices(Integer(0), distance_flag=True) {0: 0, 1: 1, 2: 2, 3: 2, 4: 1, 5: 1, 6: 2, 7: 2, 8: 2, 9: 2} - On a disconnected graph: - sage: g = 2 * graphs.RandomGNP(20, .3) sage: paths = g._backend.shortest_path_all_vertices(0) sage: all((v not in paths and g.distance(0, v) == +Infinity) or len(paths[v]) - 1 == g.distance(0, v) for v in g) True - >>> from sage.all import * >>> g = Integer(2) * graphs.RandomGNP(Integer(20), RealNumber('.3')) >>> paths = g._backend.shortest_path_all_vertices(Integer(0)) >>> all((v not in paths and g.distance(Integer(0), v) == +Infinity) or len(paths[v]) - Integer(1) == g.distance(Integer(0), v) for v in g) True 
 - shortest_path_special(x, y, exclude_vertices=None, exclude_edges=None, distance_flag=False)[source]¶
- Return the shortest path or distance from - xto- y.- This method is an extension of - shortest_path()method enabling to exclude vertices and/or edges from the search for the shortest path between- xand- y.- INPUT: - x– the starting vertex in the shortest path from- xto- y
- y– the end vertex in the shortest path from- xto- y
- exclude_vertices– iterable container (default:- None); iterable of vertices to exclude from the graph while calculating the shortest path from- xto- y
- exclude_edges– iterable container (default:- None); iterable of edges to exclude from the graph while calculating the shortest path from- xto- y
- distance_flag– boolean (default:- False); when set to- True, the shortest path distance from- xto- yis returned instead of the path
 - OUTPUT: - A list of vertices in the shortest path from - xto- yor distance from- xto- yis returned depending upon the value of parameter- distance_flag
 - EXAMPLES: - sage: G = Graph([(1, 2), (2, 3), (3, 4), (1, 5), (5, 6), (6, 7), (7, 4)]) sage: G._backend.shortest_path_special(1, 4) [1, 2, 3, 4] sage: G._backend.shortest_path_special(1, 4, exclude_vertices=[5,7]) [1, 2, 3, 4] sage: G._backend.shortest_path_special(1, 4, exclude_vertices=[2, 3]) [1, 5, 6, 7, 4] sage: G._backend.shortest_path_special(1, 4, exclude_vertices=[2], exclude_edges=[(5, 6)]) [] sage: G._backend.shortest_path_special(1, 4, exclude_vertices=[2], exclude_edges=[(2, 3)]) [1, 5, 6, 7, 4] - >>> from sage.all import * >>> G = Graph([(Integer(1), Integer(2)), (Integer(2), Integer(3)), (Integer(3), Integer(4)), (Integer(1), Integer(5)), (Integer(5), Integer(6)), (Integer(6), Integer(7)), (Integer(7), Integer(4))]) >>> G._backend.shortest_path_special(Integer(1), Integer(4)) [1, 2, 3, 4] >>> G._backend.shortest_path_special(Integer(1), Integer(4), exclude_vertices=[Integer(5),Integer(7)]) [1, 2, 3, 4] >>> G._backend.shortest_path_special(Integer(1), Integer(4), exclude_vertices=[Integer(2), Integer(3)]) [1, 5, 6, 7, 4] >>> G._backend.shortest_path_special(Integer(1), Integer(4), exclude_vertices=[Integer(2)], exclude_edges=[(Integer(5), Integer(6))]) [] >>> G._backend.shortest_path_special(Integer(1), Integer(4), exclude_vertices=[Integer(2)], exclude_edges=[(Integer(2), Integer(3))]) [1, 5, 6, 7, 4] 
 - strongly_connected_component_containing_vertex(v)[source]¶
- Return the strongly connected component containing the given vertex. - INPUT: - v– a vertex
 - EXAMPLES: - The digraph obtained from the - PetersenGraphhas an unique strongly connected component:- sage: g = DiGraph(graphs.PetersenGraph()) sage: g.strongly_connected_component_containing_vertex(0) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] - >>> from sage.all import * >>> g = DiGraph(graphs.PetersenGraph()) >>> g.strongly_connected_component_containing_vertex(Integer(0)) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] - In the Butterfly DiGraph, each vertex is a strongly connected component: - sage: g = digraphs.ButterflyGraph(3) sage: all([v] == g.strongly_connected_component_containing_vertex(v) for v in g) True - >>> from sage.all import * >>> g = digraphs.ButterflyGraph(Integer(3)) >>> all([v] == g.strongly_connected_component_containing_vertex(v) for v in g) True 
 - subgraph_given_vertices(other, vertices)[source]¶
- Initialize - otherto be the subgraph of- selfwith given vertices.- INPUT: - other– a (mutable) subclass of- CGraphBackend
- vertices– list of vertex labels
 - Note - otheris assumed to be the empty graph.- EXAMPLES: - Make a dense copy: - sage: G = sage.graphs.base.dense_graph.DenseGraphBackend(9, directed=True) sage: G.loops(True) sage: G.add_edges([[0,1], [1,2], [2,3], [3,4], [4,5], [5,6], [7,8], [3,3]], True) sage: H = sage.graphs.base.dense_graph.DenseGraphBackend(0, directed=True) sage: H.loops(True) sage: G.subgraph_given_vertices(H, range(9)) sage: list(H.iterator_out_edges(list(range(9)), False)) == list(G.iterator_out_edges(list(range(9)), False)) True - >>> from sage.all import * >>> G = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(9), directed=True) >>> G.loops(True) >>> G.add_edges([[Integer(0),Integer(1)], [Integer(1),Integer(2)], [Integer(2),Integer(3)], [Integer(3),Integer(4)], [Integer(4),Integer(5)], [Integer(5),Integer(6)], [Integer(7),Integer(8)], [Integer(3),Integer(3)]], True) >>> H = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(0), directed=True) >>> H.loops(True) >>> G.subgraph_given_vertices(H, range(Integer(9))) >>> list(H.iterator_out_edges(list(range(Integer(9))), False)) == list(G.iterator_out_edges(list(range(Integer(9))), False)) True - Make a sparse copy: - sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(0, directed=True) sage: H.loops(True) sage: G.subgraph_given_vertices(H, range(9)) sage: sorted(list(H.iterator_out_edges(list(range(9)), False))) == sorted(list(G.iterator_out_edges(list(range(9)), False))) True - >>> from sage.all import * >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0), directed=True) >>> H.loops(True) >>> G.subgraph_given_vertices(H, range(Integer(9))) >>> sorted(list(H.iterator_out_edges(list(range(Integer(9))), False))) == sorted(list(G.iterator_out_edges(list(range(Integer(9))), False))) True - Initialize a proper subgraph: - sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(0, directed=True) sage: H.loops(True) sage: G.subgraph_given_vertices(H, [2,3,4,5]) sage: list(H.iterator_out_edges(list(range(9)), False)) [(2, 3), (3, 3), (3, 4), (4, 5)] - >>> from sage.all import * >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0), directed=True) >>> H.loops(True) >>> G.subgraph_given_vertices(H, [Integer(2),Integer(3),Integer(4),Integer(5)]) >>> list(H.iterator_out_edges(list(range(Integer(9))), False)) [(2, 3), (3, 3), (3, 4), (4, 5)] - Loops are removed, if the other graph does not allow loops: - sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(0, directed=True) sage: H.loops(False) sage: G.subgraph_given_vertices(H, [2,3,4,5]) sage: list(H.iterator_out_edges(list(range(9)), False)) [(2, 3), (3, 4), (4, 5)] - >>> from sage.all import * >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0), directed=True) >>> H.loops(False) >>> G.subgraph_given_vertices(H, [Integer(2),Integer(3),Integer(4),Integer(5)]) >>> list(H.iterator_out_edges(list(range(Integer(9))), False)) [(2, 3), (3, 4), (4, 5)] - Multiple edges and labels are copied: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(4, directed=False) sage: G.multiple_edges(True) sage: G.add_edges([[0,1,'a'], [1,2,'b'], [2,3,'c'], [0,1,'d']], False) sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(0, directed=False) sage: H.multiple_edges(True) sage: G.subgraph_given_vertices(H, [0,1,2]) sage: list(H.iterator_edges(list(range(4)), True)) [(0, 1, 'a'), (0, 1, 'd'), (1, 2, 'b')] - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(4), directed=False) >>> G.multiple_edges(True) >>> G.add_edges([[Integer(0),Integer(1),'a'], [Integer(1),Integer(2),'b'], [Integer(2),Integer(3),'c'], [Integer(0),Integer(1),'d']], False) >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0), directed=False) >>> H.multiple_edges(True) >>> G.subgraph_given_vertices(H, [Integer(0),Integer(1),Integer(2)]) >>> list(H.iterator_edges(list(range(Integer(4))), True)) [(0, 1, 'a'), (0, 1, 'd'), (1, 2, 'b')] - Multiple edges are removed, if the other graph does not allow them: - sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(0, directed=False) sage: H.multiple_edges(False) sage: G.subgraph_given_vertices(H, [0,1,2]) sage: list(H.iterator_edges(list(range(4)), True)) [(0, 1, 'd'), (1, 2, 'b')] - >>> from sage.all import * >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0), directed=False) >>> H.multiple_edges(False) >>> G.subgraph_given_vertices(H, [Integer(0),Integer(1),Integer(2)]) >>> list(H.iterator_edges(list(range(Integer(4))), True)) [(0, 1, 'd'), (1, 2, 'b')] - Labels are removed, if the other graph does not allow them: - sage: H = sage.graphs.base.dense_graph.DenseGraphBackend(0, directed=False) sage: G.subgraph_given_vertices(H, [0,1,2]) sage: list(H.iterator_edges(list(range(4)), True)) [(0, 1, None), (1, 2, None)] - >>> from sage.all import * >>> H = sage.graphs.base.dense_graph.DenseGraphBackend(Integer(0), directed=False) >>> G.subgraph_given_vertices(H, [Integer(0),Integer(1),Integer(2)]) >>> list(H.iterator_edges(list(range(Integer(4))), True)) [(0, 1, None), (1, 2, None)] - A directed subgraph of an undirected graph is taken by initializing with edges in both directions: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(4, directed=True) sage: G.loops(True) sage: G.multiple_edges(True) sage: G.add_edges([[0,1,'a'], [1,2,'b'], [2,3,'c'], [0,1,'d'], [2,2,'e']], False) sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(0, directed=True) sage: H.multiple_edges(True) sage: H.loops(True) sage: G.subgraph_given_vertices(H, [0,1,2]) sage: list(H.iterator_out_edges(list(range(4)), True)) [(0, 1, 'a'), (0, 1, 'd'), (1, 0, 'a'), (1, 0, 'd'), (1, 2, 'b'), (2, 1, 'b'), (2, 2, 'e')] - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(4), directed=True) >>> G.loops(True) >>> G.multiple_edges(True) >>> G.add_edges([[Integer(0),Integer(1),'a'], [Integer(1),Integer(2),'b'], [Integer(2),Integer(3),'c'], [Integer(0),Integer(1),'d'], [Integer(2),Integer(2),'e']], False) >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0), directed=True) >>> H.multiple_edges(True) >>> H.loops(True) >>> G.subgraph_given_vertices(H, [Integer(0),Integer(1),Integer(2)]) >>> list(H.iterator_out_edges(list(range(Integer(4))), True)) [(0, 1, 'a'), (0, 1, 'd'), (1, 0, 'a'), (1, 0, 'd'), (1, 2, 'b'), (2, 1, 'b'), (2, 2, 'e')] - An undirected subgraph of a directeed graph is not defined: - sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(4, directed=True) sage: G.add_edges([[0,1,'a'], [1,2,'b'], [2,3,'c']], False) sage: H = sage.graphs.base.sparse_graph.SparseGraphBackend(0, directed=False) sage: G.subgraph_given_vertices(H, [0,1,2]) Traceback (most recent call last): ... ValueError: cannot obtain an undirected subgraph of a directed graph - >>> from sage.all import * >>> G = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(4), directed=True) >>> G.add_edges([[Integer(0),Integer(1),'a'], [Integer(1),Integer(2),'b'], [Integer(2),Integer(3),'c']], False) >>> H = sage.graphs.base.sparse_graph.SparseGraphBackend(Integer(0), directed=False) >>> G.subgraph_given_vertices(H, [Integer(0),Integer(1),Integer(2)]) Traceback (most recent call last): ... ValueError: cannot obtain an undirected subgraph of a directed graph 
 
- class sage.graphs.base.c_graph.Search_iterator[source]¶
- Bases: - object- An iterator for traversing a (di)graph. - This class is commonly used to perform a depth-first or breadth-first search. The class does not build all at once in memory the whole list of visited vertices. The class maintains the following variables: - graph– a graph whose vertices are to be iterated over
- direction– integer; this determines the position at which vertices to be visited are removed from the list. For breadth-first search (BFS), element removal follow a first-in first-out (FIFO) protocol, as signified by the value- direction=0. We use a queue to maintain the list of vertices to visit in this case. For depth-first search (DFS), element removal follow a last-in first-out (LIFO) protocol, as signified by the value- direction=-1. In this case, we use a stack to maintain the list of vertices to visit.
- stack– list of vertices to visit, used only when- direction=-1
- queue– a queue of vertices to visit, used only when- direction=0
- seen– list of vertices that are already visited
- test_out– boolean; whether we want to consider the out-neighbors of the graph to be traversed. For undirected graphs, we consider both the in- and out-neighbors. However, for digraphs we only traverse along out-neighbors.
- test_in– boolean; whether we want to consider the in-neighbors of the graph to be traversed. For undirected graphs, we consider both the in- and out-neighbors.
 - EXAMPLES: - sage: g = graphs.PetersenGraph() sage: list(g.breadth_first_search(0)) [0, 1, 4, 5, 2, 6, 3, 9, 7, 8] - >>> from sage.all import * >>> g = graphs.PetersenGraph() >>> list(g.breadth_first_search(Integer(0))) [0, 1, 4, 5, 2, 6, 3, 9, 7, 8]