The flipping puzzle on a graph

TL;DR

This paper analyzes a graph-based flipping puzzle, providing a complete characterization of when one configuration can be transformed into another through a sequence of moves, based on the structure of the graph.

Contribution

It offers a complete solution to the puzzle on connected graphs containing an induced path, detailing the conditions for configuration reachability.

Findings

Characterization of configuration reachability on such graphs

Explicit move sequences for certain configurations

Conditions based on the graph's induced path structure

Abstract

Let $S$ be a connected graph which contains an induced path of $n-1$ vertices, where $n$ is the order of $S.$ We consider a puzzle on $S$. A configuration of the puzzle is simply an $n$-dimensional column vector over $\{0, 1\}$ with coordinates of the vector indexed by the vertex set $S$. For each configuration $u$ with a coordinate $u_s=1$, there exists a move that sends $u$ to the new configuration which flips the entries of the coordinates adjacent to $s$ in $u.$ We completely determine if one configuration can move to another in a sequence of finite steps.