Domination games on graphs

This study focuses on two games played on graphs, called the Domination Game and the Total Domination Game. These games are played by two players, Dominator and Staller, who take turns selecting vertices from the graph. In the Domination Game, each selected vertex must dominate at least one previously undominated vertex. The goal of the Dominator is to end the game in as few moves as possible, while the Staller aims to prolong the game. The game ends when no further moves can be made and all vertices in the graph are dominated. The Game Domination Number measures the number of moves made in an optimally played game. The Total Domination Game has a similar structure, but a stricter rule applies: Each selected vertex must dominate at least one vertex, other than itself, that has not been dominated in any way by the previously selected vertices. The Game Total Domination Number represents the number of moves required to end the game based on this rule. In the study, the game domination and game total domination numbers have been analyzed on various families of graphs, existing results from the literature have been presented, and the behavior of these games on different graph structures has been examined. Keywords: Domination game, Game domination number, Graphs