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 $γ_t^{\rm i}(G)$. 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.

Figures (7)

• Figure 1: A graph $G$
• Figure 2: The graph $G_k$ in the family $\mathcal{G}$
• Figure 3: The graph $F_k = (K_k \,\square\, K_2) - u_kv_k$
• Figure 4: The graphs $B_4$ and $J_4$
• Figure 5: The tree $T_k$ in the proof of Proposition \ref{['prop:subdivide-star']}

Theorems & Definitions (18)

• Lemma 1
• Proposition 1
• Proposition 2
• Theorem 1
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Theorem 2
• Proposition 7