Game total domination subdivision number of a graph

Abstract

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγt(G) is the minimum cardinality of a total dominating set of G. The game total domination subdivision number of a graph G is defined by the following game. Two players 𝒟 and 𝒜, 𝒟 playing first, alternately mark or subdivide an edge of G which is not yet marked nor subdivided. The game ends when all the edges of G are marked or subdivided and results in a new graph G′. The purpose of 𝒟 is to minimize the total domination number γt(G′) of G′ while 𝒜 tries to maximize it. If both 𝒜 and 𝒟 play according to their optimal strategies, γt(G′) is well defined. We call this number the game total domination subdivision number of G and denote it by γgt(G). In this paper we initiate the study of the game total domination subdivision number of a graph and present some (sharp) bounds for this parameter.