Vol. 11, No. 1, 2018

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Merging peg solitaire on graphs

John Engbers and Ryan Weber

Vol. 11 (2018), No. 1, 53–66
DOI: 10.2140/involve.2018.11.53

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.

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peg solitaire, games on graphs, graph theory
Mathematical Subject Classification 2010
Primary: 05C57
Received: 14 February 2016
Revised: 5 August 2016
Accepted: 7 August 2016
Published: 17 July 2017

Communicated by Anant Godbole
John Engbers
Department of Mathematics, Statistics and Computer Science
Marquette University
Milwaukee, WI 53201
United States
Ryan Weber
Department of Mathematics, Statistics and Computer Science
Marquette University
Milwaukee, WI 53201
United States