Fill in the maximum matching on the graph from the end.

https://gyazo.com/8f005eab033d686a1a4adbdafc70dcec

When considering the maximum matching on a graph, a point with rank 1 can be considered to be included in the maximum matching

The edge containing the point with rank 1 is called the "end edge", and the edge containing the other side of the edge is called the "next edge".

The next edge is zero or more in the general graph

When you choose a certain maximum matching

1: If neither the end edge nor the next edge is selected (or the next edge does not exist), then selecting the end edge increases the number of matches by 1

In other words, it was never a maximum match to begin with.

2: When an edge is unselected and the next edge is selected, swapping selections does not increase or decrease

Thus, the exchanged ones are also the maximum matching

3: Neither the end edge nor the next edge is selected for matching.

From 1 and 2, some of the largest matches always have end edges selected.

So you can select an edge and remove it.

In restricted graphs, this "edge of a blade removal" results in a new [edge of a blade

It does not hold true for general graphs.

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