Paths in Directed Acyclic Graphs¶
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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
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class
sage.combinat.graph_path.GraphPaths_all(g)¶ Bases:
sage.combinat.combinat.CombinatorialClass,sage.combinat.graph_path.GraphPaths_commonEXAMPLES:
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
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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
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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)]
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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]]
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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)]
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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]]
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paths()¶ Return 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
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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]]
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class
sage.combinat.graph_path.GraphPaths_s(g, source)¶ Bases:
sage.combinat.combinat.CombinatorialClass,sage.combinat.graph_path.GraphPaths_common-
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]]
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class
sage.combinat.graph_path.GraphPaths_st(g, source, target)¶ Bases:
sage.combinat.combinat.CombinatorialClass,sage.combinat.graph_path.GraphPaths_commonEXAMPLES:
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
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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]]
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class
sage.combinat.graph_path.GraphPaths_t(g, target)¶ Bases:
sage.combinat.combinat.CombinatorialClass,sage.combinat.graph_path.GraphPaths_common-
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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