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# has_eulerian_path

has_eulerian_path(G, source=None)

  Return True iff `G` has an Eulerian path.

  An Eulerian path is a path in a graph which uses each edge of a graph exactly once. If `source` is specified, then this function checks whether an Eulerian path that starts at node `source` exists.

  A directed graph has an Eulerian path iff:

    - at most one vertex has out_degree - in_degree = 1,

    - at most one vertex has in_degree - out_degree = 1,

    - every other vertex has equal in_degree and out_degree,

    - and all of its vertices belong to a single connected component of the underlying undirected graph.

  If `source` is not None, an Eulerian path starting at `source` exists if no other node has out_degree - in_degree = 1. This is equivalent to either there exists an Eulerian circuit or `source` has out_degree - in_degree = 1 and the conditions above hold.

  An undirected graph has an Eulerian path iff:

    - exactly zero or two vertices have odd degree,

    - and all of its vertices belong to a single connected component.

  If `source` is not None, an Eulerian path starting at `source` exists if either there exists an Eulerian circuit or `source` has an odd degree and the conditions above hold.

  Graphs with isolated vertices (i.e. vertices with zero degree) are not considered to have an Eulerian path. Therefore, if the graph is not connected (or not strongly connected, for directed graphs), this function returns False.

  Parameters:
    G NetworkX Graph

      The graph to find an euler path in.

    source node, optional

      Starting node for path.

  Returns:
    Bool True if G has an Eulerian path.

  See also

  `is_eulerian`
  `eulerian_path`

  Examples

  If you prefer to allow graphs with isolated vertices to have Eulerian path, you can first remove such vertices and then call `has_eulerian_path` as below example shows.

      >>> G = nx.Graph([(0, 1), (1, 2), (0, 2)])
      >>> G.add_node(3)
      >>> nx.has_eulerian_path(G)
      False

      >>> G.remove_nodes_from(list(nx.isolates(G)))
      >>> nx.has_eulerian_path(G)
      True