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.

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 is to have a single peg remaining. In a 2011 paper, this game is generalized to graphs. In this paper, we consider a variation in which each peg must be jumped twice in order to be removed. For this variation, we consider the solvability of several graph families. For our major results, we characterize solvable joins of graphs and show that the Cartesian product of solvable graphs is likewise solvable.

There have been several papers on the subject of traditional peg solitaire on different boards. However, in this paper we consider a generalization of the game to arbitrary boards. These boards are treated as graphs in the combinatorial sense. We present necessary and sufficient conditions for the solvability of several well-known families of graphs. In the major result of this paper, we show that the cartesian product of two solvable graphs is likewise solvable. Several related results are also presented. Finally, several open problems related to this study are given.

In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families.

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. When the game is played between two players it is called duotaire. In this paper, we consider two variations of peg duotaire on graphs. In the first variation, the last player to remove a peg wins. Inspired by the work of Slater, we also investigate a variation in which one player tries to maximize the number of pegs at the end of the game while their opponent seeks to minimize this number. For both variations, we give explicit strategies for several families of graphs. Finally, we give a number of open problems as possible avenues for future research.

In a paper by Beeler and Hoilman, the traditional game of peg solitaire is generalized to graphs in the combinatorial sense. One of the important open problems in this paper was to classify solvable trees. In this thesis, we will give necessary and sufficient conditions for the solvability for all trees with diameter four. We also give the maximum number of pegs that can be left on such a graph under the restriction that we jump whenever possible.