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# 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)

  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

  `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)

  Bases: `sage.combinat.combinat.CombinatorialClass`, `sage.combinat.graph_path.GraphPaths_common`

  TESTS:

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

  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

  `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)

  Bases: `sage.combinat.combinat.CombinatorialClass`, `sage.combinat.graph_path.GraphPaths_common`

  TESTS:

      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 == loads(dumps(p))
      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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