Trees with Equal Total Domination and Game Total Domination Numbers

TL;DR

This paper investigates the total domination game on graphs, proving that for forests with no isolated vertices, the game total domination number when Dominator starts is at most when Staller starts, and characterizes trees where these numbers are equal.

Contribution

It establishes a bound on the game total domination numbers for forests and characterizes trees with equal total domination and game total domination numbers.

Findings

For forests with no isolated vertices, γ_tg(G) ≤ γ_tg'(G).

Characterization of trees with equal total domination and game total domination numbers.

Abstract

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set $S$ of $G$ in which every vertex is totally dominated by a vertex in $S$. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game total domination number, $\gamma_{\rm tg}(G)$, (respectively, Staller-start game total domination number, $\gamma_{\rm tg}'(G)$) of $G$ is the number of vertices chosen when Dominator (respectively, Staller) starts the game and both players play optimally. For general graphs $G$, sometimes $\gamma_{\rm tg}(G) > \gamma_{\rm tg}'(G)$. We show that if $G$ is a forest with no isolated vertex, then $\gamma_{\rm tg}(G) \le \gamma_{\rm tg}'(G)$. Using this result, we characterize the trees with equal total domination and game total domination number.