Abstract
Let G = (V,E) be a connected graph. We say a set U ⊆ V is P3-convex if every vertex of V\U has at most one neighbor in U. For any set W ⊆ V, we use σ(W) to denote the unique minimal P3-convex set of vertices that contains W. Two players play the connected P3-game on a graph by alternately selecting unmarked vertices. At the start of the game all vertices are unmarked, and the set of marked vertices, denoted by M, is empty. A move consists of marking a previously unmarked vertex v, so that M is updated to M0 = σ(M ∪{v}), and such a move is legal only when M0 is connected. When it is a player’s turn to move, he loses the game if all vertices are marked. In this thesis, we consider the connected P3-game on grids and tori, and give closed form formulae of their corresponding Grundy values. We also consider the related Pr-game, with r ≥ 3, that is played on a path or a cycle.