Merging peg solitaire on graphs

Abstract

Peg solitaire has recently been generalized to graphs. Here, pegs start on all but one of the vertices in a graph. A move takes pegs on adjacent vertices $x$ and $y$, with $y$ also adjacent to a hole on vertex $z$, and jumps the peg on $x$ over the peg on $y$ to $z$, removing the peg on $y$. The goal of the game is to reduce the number of pegs to one.

We introduce the game merging peg solitaire on graphs, where a move takes pegs on vertices $x$ and $z$ (with a hole on $y$) and merges them to a single peg on $y$. When can a configuration on a graph, consisting of pegs on all vertices but one, be reduced to a configuration with only a single peg? We give results for a number of graph classes, including stars, paths, cycles, complete bipartite graphs, and some caterpillars.