Game total domination for cyclic bipartite graphs

Abstract

Let G=(V,E) be a graph. A vertex u in G totally dominates a vertex v if u is adjacent to v in G. The total domination game played on G consists of two players, named Dominator and Staller, who alternately take turns choosing vertices of G such that each chosen vertex totally dominates at least one vertex not totally dominated by the vertices previously chosen. Dominator wishes to totally dominate the graph as fast as possible, while Staller wishes to delay the process as much as possible. The game total domination number γtg(G) (resp. the Staller-start game total domination number γtg′(G)) of G is the number of vertices chosen when Dominator starts the game (resp. when Staller starts the game) and both players play optimally. In this paper, we determine the exact value of γtg(G) and γtg′(G) when G is a cyclic bipartite graph.