Further Progress on the Heredity of the Game Domination Number

Abstract

The domination game is a two-player game played on a finite, undirected graph G. During the game, the players alternately choose a vertex of G such that each chosen vertex dominates at least one previously undominated vertex. One player, called Dominator, tries to finish the game within few moves, while the second player, Staller, tries to make it last for as long as possible. The game domination number \({\gamma _g}(G)\) is the total number of moves in the game when Dominator starts and both players play optimally. The Staller start game domination number \({\gamma _g'}(G)\) is defined similarly when Staller starts the game. The behaviour of the game domination number on the removal of a vertex and an edge so as that no heredity is possible, in contrast with what is happening for domination. In this paper we consider the special case of no-minus-graphs.