Reversible Peg Solitaire on Graphs

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

Merging Peg Solitaire in Graphs

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 ony 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

Making graphs solvable in peg solitaire

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire has been considered on quite a few classes of graphs. Beeler and Gray introduced the natural idea of adding edges to make an unsolvable graph solvable. Recently, the graph invariant ms(G), which is the minimal number of additional edges needed to make G solvable, has been introduced and investigated on banana trees by the authors. In this article, we determine ms(G) for several families of unsolvable graphs. Furthermore, we provide some general results for this number of Hamiltonian graphs and graphs obtained via binary graph operations

Peg Solitaire on Cartesian Products of Graphs

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the "most" unsolvable graphs

Path-Stick Solitaire on Graphs

In 2011, Beeler and Hoilman generalised the game of peg solitaire to arbitrary connected graphs. Since then, peg solitaire and related games have been considered on many graph classes. In this paper, we introduce a variant of the game peg solitaire, called path-stick solitaire, which is played with sticks in edges instead of pegs in vertices. We prove several analogues to peg solitaire results for that game, mainly regarding different graph classes. Furthermore, we characterise, with very few exceptions, path-stick-solvable joins of graphs and provide some possible future research questions

Peg Solitaire on Graphs In Which We Allow Merging and Jumping

Peg solitaire is a game in which pegs are placed in every hole but one and the player jumps over pegs along rows or columns to remove them. Usually, the goal of the player is to leave only one peg. In a 2011 paper, this game is generalized to graphs. In this thesis, we consider a variation of peg solitaire on graphs in which pegs can be removed either by jumping them or merging them together. To motivate this, we survey some of the previous papers in the literature. We then determine the solvability of several classes of graphs including stars and double stars, caterpillars, trees of small diameter, particularly four and five, and articulated caterpillars. We conclude this thesis with several open problems related to this study

グラフ上のペグソリティアの計算複雑さ

　オセロやチェスをはじめとしたゲーム・パズルは古くから遊ばれており、その計算複雑さや必勝性等についても盛んに研究が行われている。本論文ではゲーム・パズルの一種であるペグソリティアをグラフ上に一般化したグラフ上のペグソリティアの計算複雑さ及び可解性について研究を行う。　ペグソリティアとは格子状の盤上のペグを定められたルールで取り除いていき最終的に1個にすることを目的とする古くから遊ばれているパズルである。ペグソリティアの解法、可解性については様々な研究が行われており、またその計算量については与えられた盤面でのペグソリティアが解を持つか否かを判定する問題はNP完全であることが知られている。ペグソリティアは格子状の盤面の上でペグを動かすが、このペグソリティアを一般のグラフ上に拡張したグラフ上のペグソリティアも考案され、研究が行われている。しかしグラフ上のペグソリティアの計算困難性は証明されていなかった。本研究ではこのグラフ上のペグソリティアにおいて、グラフと初期状態で空となる1個の頂点が与えられたときペグを一つにできるか否かという問題がNP完全であることを証明した。入次数1出次数2の頂点もしくは入次数2出次数1の頂点のみで構成される平面有向グラフに対するハミルトン閉路問題からの帰着により、証明をおこなった。　また、グラフ上のペグソリティアでは標準的なペグソリティアを表すことができない。そこで標準的なペグソリティアを表せるような定義をしたグラフ上のペグソリティアも考案されており、これを一般化ペグソリティアと呼び通常のグラフ上のペグソリティアと区別する。本研究ではこの一般化ペグソリティアにおいて、パスとサイクルの直積でできたグラフの可解性についても研究をおこなった。その結果C3□Pn (n ≧ 3)、C4□Pn (n ≧ 3)、Cn□P3 (n ≧ 3)、C5□P4、C2n□P5 (n ≧ 2)、C2n□P2m (n, m ≧ 2)が解を持つことを証明した。電気通信大学201