Minimal and irreducible links in the Shannon game

TL;DR

This paper analyzes weak and strong links in the Shannon game, introduces methods to find all such links of small size, and explores their reducibility, with applications to Hex game analysis.

Contribution

It presents a comprehensive method to identify all minimal links of given sizes and introduces a new reduction called the 'short-cut' for analyzing these links.

Findings

Complete sets of irreducible weak links on up to 11 vertices

Complete sets of strong links on up to 10 vertices

Introduction of the 'short-cut' reduction method

Abstract

We discuss weak and strong links (`virtual connections') in the Shannon game. General properties of these links are discussed, leading to a method to find all links of given size by a suitably pruned exhaustive search. This is applied to links on graphs of up to 11 vertices. We discuss the concept of reducibility of such links. Three simple reductions are considered, including one, the `short-cut', not previously described. The complete sets of irreducible weak links on up to 11 vertices and strong links on up to 10 vertices are presented. Some applications to the analysis of Hex are noted.