# Paths in Directed Acyclic Graphs¶

sage.combinat.graph_path.GraphPaths(g, source=None, target=None)

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)
Traceback (most recent call last):
...
TypeError: g must be a directed acyclic graph

class sage.combinat.graph_path.GraphPaths_all(g)

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

list()

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

class sage.combinat.graph_path.GraphPaths_common
incoming_edges(v)

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)]

incoming_paths(v)

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]]

outgoing_edges(v)

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)]

outgoing_paths(v)

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]]

paths()

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

paths_from_source_to_target(source, target)

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]]

class sage.combinat.graph_path.GraphPaths_s(g, source)

TESTS:

sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G, 4)
True

list()

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]]

class sage.combinat.graph_path.GraphPaths_st(g, source, target)

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

list()

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]]

class sage.combinat.graph_path.GraphPaths_t(g, target)

TESTS:

sage: G = DiGraph({1:[2,2,3], 2:[3,4], 3:[4], 4:[5,5]}, multiedges=True)
sage: p = GraphPaths(G, target=4)
True

list()

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]]


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