Reversible peg solitaire on graphs

TL;DR

This paper investigates reversible peg solitaire on graphs, proving which graph classes are solvable and classifying initial and final configurations, expanding understanding of the game's solvability conditions.

Contribution

It characterizes the solvability of reversible peg solitaire on various graph classes and classifies feasible starting and ending positions.

Findings

All non-star graphs with a vertex of degree at least three are solvable.

Cycles and paths with a number of vertices divisible by 2 or 3 are solvable.

Other graphs are not solvable.

Abstract

The game of peg solitaire on graphs was introduced by Beeler and Hoilman in 2011. In this game, pegs are initially placed on all but one vertex of a graph $G$. If $xyz$ forms a path in $G$ and there are pegs on vertices $x$ and $y$ but not $z$, then a {\em jump} places a peg on $z$ and removes the pegs from $x$ and $y$. A graph is called solvable if, for some configuration of pegs occupying all but one vertex, some sequence of jumps leaves a single peg. We study the game of {\em reversible peg solitaire}, where there are again initially pegs on all but one vertex, but now both jumps and unjumps (the reversal of a jump) are allowed. We show that in this game all non-star graphs that contain a vertex of degree at least three are solvable, that cycles and paths on $n$ vertices, where $n$ is divisible by $2$ or $3$, are solvable, and that all other graphs are not solvable. We also classify the possible starting hole and ending peg positions for solvable graphs.