Connect Four and Graph Decomposition

TL;DR

This paper introduces a graph decomposition method called standard decomposition, relates it to Connect Four game decompositions, and proves that all such decompositions can be generated efficiently in polynomial time.

Contribution

It establishes a polynomial-time algorithm for generating all standard graph decompositions and connects this to Connect Four game decompositions.

Findings

All standard decompositions can be generated in polynomial time.

Connect Four decompositions are a special case of standard graph decompositions.

The method provides a new graph-theoretic perspective on Connect Four.

Abstract

We introduce the standard decomposition, a way of decomposing a labeled graph into a sum of certain labeled subgraphs. We motivate this graph-theoretic concept by relating it to Connect Four decompositions of standard sets. We prove that all standard decompositions can be generated in polynomial time, which implies that all Connect Four decompositions can be generated in polynomial time.