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:

GNetworkX Graph

The graph to find an euler path in.

sourcenode, optional

Starting node for path.

Returns:

BoolTrue 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