Biased Domination Games

Abstract

A Domination game on a graph G is a game of two players, called Dominator and Staller, on a graph. The players take turns to perform a move by choosing a vertex in the graph. Vertices in the closed neighborhood of a chosen vertex are said to be dominated. A move u is legal if it creates at least one new dominated vertex. The game is ended when all vertices in the graph are dominated. Dominator tries to end the game as soon as possible, while Staller tries to prolong the game. In the domination game, if Dominator starts the game, this game is said to be Game 1. Otherwise, it is said to be Game 2. If both players play optimally in a domination game on a graph G, the number of moves when the game is ended is called the game domination numbers. In this research, we introduce an extended version of a domination game on a graph, called a biased domination game, in which Dominator and Staller play more than one move in each turn. Similarly, we define the biased game domination number as the number of moves in an ended biased domination game which both players play with optimal strategies. We study relations of biased game domination numbers between various games. In addition, we study two special types of moves, called minimal moves and maximal moves. Some properties of the biased game domination numbers on a graph where the special moves are always available is studied. Lastly, the biased game domination numbers of powers of a cycle are explicitly computed, together with optimal strategies using a special move.