Figure (b) above is an example of a perfect matching.

In the above figure, only part (b) shows a perfect matching.

The problems associated with the need for perfect matching can be avoided by one further design change.

It was known before that the decision problem "Is there a perfect matching for a given bipartite graph?"

Because must contain an even number of vertices, a perfect matching exists.

We want to find a perfect matching with minimum cost.

We are done if M is a perfect matching.

One can then use polynomial identity testing to find whether contains a perfect matching.

The excess is easily peeled away to leave a perfect matching join.

This is again another perfect matching of these eight points.