# Does Tutte's theorem apply for simple graphs only

Tutte's theorem as stated on wikipedia says:

A graph, $G = (V, E)$, has a perfect matching if and only if for every subset $U$ of $V$, the subgraph induced by $V − U$ has at most $|U|$ connected components with an odd number of vertices.

Is the graph $G$ assumed to be simple? If not, why does Andersen's proof of Tutte's theorem here begin by considering only simple graphs?

• A graph with multiple edges has a perfect matching if and only if the underlying simple graph has a perefect matching. – Chris Godsil Feb 17 '16 at 5:19

The statement of Tutte's theorem does not really mention edges. So suppose there are possibly loops and multiple edges between vertices. Then $V-U$ has exactly the same number of connected components as the graph you'd get by removing the extra edges and loops because deleting a vertex from a graph removes all the edges and loops do not do anything for connectivity.