Graph Minors

Take a graph and remove some edges. Then contract some edges so that their ends become fused. What you get is a minor of the original graph.

There are some interesting results about graph minors. K5 is the graph made of 5 vertices which are all connected (requiring 10 edges). K3,3 is two sets of three vertices, each vertex in one set connected to each vertex in the other (requires 9 edges).

An even deeper result says that in any infinite sequence of finite graphs you can always find one graph that is a minor of another.

Deciding whether G1 is a minor of G2 is NpHard.

References:

This paper gives some concrete algorithmic implications of the work:


CategoryMath


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