Indicated total domination game

Abstract

A vertex u in a graph G totally dominates a vertex v if u is adjacent to v in G. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The indicated total domination game is played on a graph G by two players, Dominator and Staller, who take turns making a move. In each of his moves, Dominator indicates a vertex v of the graph that has not been totally dominated in the previous moves, and Staller chooses (or selects) any vertex adjacent to v that has not yet been played, and adds it to a set D that is being built during the game. The game ends when every vertex is totally dominated, that is, when D is a total dominating set of G. The goal of Dominator is to minimize the size of D, while Staller wants just the opposite. Providing that both players are playing optimally with respect to their goals, the size of the resulting set D is the indicated total domination number of G, denoted by γ i t (G). In this paper we present several results on indicated total domination game. Among other results we prove that the indicated total domination number of a graph is bounded below by the well studied upper total domination number.