Theory of Combinatorial Games in Graphs

This book offers a comprehensive introduction to the field of combinatorial games, with a contemporary focus on games played on graphs. It provides a clear, structured tour of the major classes of combinatorial games (normal, misère, impartial, partizan, and positional), illustrated throughout with graph-based examples. The book is divided into three parts. Part I presents the fundamental theoretical foundations of combinatorial game theory. Part II explores their applications to recently studied games on graphs, and Part III provides a summary of the theory of partizan games in the normal variant. Readers will find coverage of the Sprague–Grundy theory for impartial games, extremal combinatorics in game settings, computational complexity of games, convexity games on graphs, domination games, cops-and-robber games, as well as Conway’s theory of partizan games and surreal numbers. Beyond its introductory material, the book also brings together several active research topics that are typically scattered across the literature, such as graph coloring games, graph convexity games, and connectivity games. Although primarily designed for undergraduate students, the book’s more advanced results will also be valuable to graduate students and researchers working in the area.