Completely symmetric configurations for sigma-games on grid graphs

TL;DR

This paper characterizes when symmetric configurations can be reached in sigma-games on grid graphs across various dimensions, providing complete answers in many cases and highlighting open problems.

Contribution

It offers a comprehensive analysis of symmetric configurations in sigma-games on grid graphs, including new results for dimensions two and higher, and identifies unresolved cases.

Findings

Complete characterization in 2D for all symmetric configurations.

Results for sigma^+ -game in dimensions ≥3.

Partial results for sigma^- -game with even sizes.

Abstract

The paper deals with sigma-games on grid graphs (in dimension 2 and more) and conditions under which any completely symmetric configuration of lit vertices can be reached -- in particular the completely lit configuration -- when starting with the all-unlit configuration. The answer is complete in dimension 2. In dimension greater than or equal to 3, the answer is complete for the sigma^+ -game, and for the sigma^- -game if at least one of the sizes is even. The case sigma^-, dimension greater than or equal to 3 and all sizes odd remains open.