Document Type

Article

Language

eng

Format of Original

6 p.

Publication Date

11-2015

Publisher

Elsevier

Source Publication

Discrete Mathematics

Source ISSN

0012-365X

Original Item ID

doi: 10.1016/j.disc.2015.05.012

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 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 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.

Comments

Accepted version. Discrete Mathematics, Vol. 338, No. 11 (November 2015): 2014-2019. DOI. © 2015 Elsevier. Used with permission.

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