Returns the combinatorial class of paths in the directed acyclic graph g.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
If source and target are not given, then the returned class contains all paths (including trivial paths containing only one vertex).
sage: p = GraphPaths(G); p
Paths in Multi-digraph on 5 vertices
sage: p.cardinality()
37
sage: p.random_element()
[1, 2, 3, 4, 5]
If the source is specified, then the returned class contains all of the paths starting at the vertex source (including the trivial path).
sage: p = GraphPaths(G, source=3); p
Paths in Multi-digraph on 5 vertices starting at 3
sage: p.list()
[[3], [3, 4], [3, 4, 5], [3, 4, 5]]
If the target is specified, then the returned class contains all of the paths ending at the vertex target (including the trivial path).
sage: p = GraphPaths(G, target=3); p
Paths in Multi-digraph on 5 vertices ending at 3
sage: p.cardinality()
5
sage: p.list()
[[3], [1, 3], [2, 3], [1, 2, 3], [1, 2, 3]]
If both the target and source are specified, then the returned class contains all of the paths from source to target.
sage: p = GraphPaths(G, source=1, target=3); p
Paths in Multi-digraph on 5 vertices starting at 1 and ending at 3
sage: p.cardinality()
3
sage: p.list()
[[1, 2, 3], [1, 2, 3], [1, 3]]
Note that G must be a directed acyclic graph.
sage: G = DiGraph({1:[2,2,3,5], 2:[3,4], 3:[4], 4:[2,5,7], 5:[6]}, multiedges=True)
sage: GraphPaths(G)
...
TypeError: g must be a directed acyclic graph
Bases: sage.combinat.combinat.CombinatorialClass, sage.combinat.graph_path.GraphPaths_common
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G)
sage: p.cardinality()
37
Returns a list of the paths of self.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: len(GraphPaths(G).list())
37
Returns a list of v’s incoming edges.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G)
sage: p.incoming_edges(2)
[(1, 2, None), (1, 2, None)]
Returns a list of paths that end at v.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: gp = GraphPaths(G)
sage: gp.incoming_paths(2)
[[2], [1, 2], [1, 2]]
Returns a list of v’s outgoing edges.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G)
sage: p.outgoing_edges(2)
[(2, 3, None), (2, 4, None)]
Returns a list of the paths that start at v.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: gp = GraphPaths(G)
sage: gp.outgoing_paths(3)
[[3], [3, 4], [3, 4, 5], [3, 4, 5]]
sage: gp.outgoing_paths(2)
[[2],
[2, 3],
[2, 3, 4],
[2, 3, 4, 5],
[2, 3, 4, 5],
[2, 4],
[2, 4, 5],
[2, 4, 5]]
Returns a list of all the paths of self.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: gp = GraphPaths(G)
sage: len(gp.paths())
37
Returns a list of paths from source to target.
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: gp = GraphPaths(G)
sage: gp.paths_from_source_to_target(2,4)
[[2, 3, 4], [2, 4]]
Bases: sage.combinat.combinat.CombinatorialClass, sage.combinat.graph_path.GraphPaths_common
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G, 4)
sage: p.list()
[[4], [4, 5], [4, 5]]
Bases: sage.combinat.combinat.CombinatorialClass, sage.combinat.graph_path.GraphPaths_common
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: GraphPaths(G,1,2).cardinality()
2
sage: GraphPaths(G,1,3).cardinality()
3
sage: GraphPaths(G,1,4).cardinality()
5
sage: GraphPaths(G,1,5).cardinality()
10
sage: GraphPaths(G,2,3).cardinality()
1
sage: GraphPaths(G,2,4).cardinality()
2
sage: GraphPaths(G,2,5).cardinality()
4
sage: GraphPaths(G,3,4).cardinality()
1
sage: GraphPaths(G,3,5).cardinality()
2
sage: GraphPaths(G,4,5).cardinality()
2
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G,1,2)
sage: p.list()
[[1, 2], [1, 2]]
Bases: sage.combinat.combinat.CombinatorialClass, sage.combinat.graph_path.GraphPaths_common
EXAMPLES:
sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G, target=4)
sage: p.list()
[[4],
[2, 4],
[1, 2, 4],
[1, 2, 4],
[3, 4],
[1, 3, 4],
[2, 3, 4],
[1, 2, 3, 4],
[1, 2, 3, 4]]