On the game total domination number

TL;DR

This paper investigates the game total domination number in graphs, establishing an improved upper bound for graphs without isolated vertices or edges, advancing understanding of optimal strategies in this combinatorial game.

Contribution

The paper proves a new upper bound of 11/14 times the number of vertices for the game total domination number in specific graphs, improving previous bounds.

Findings

Proved that γ_{tg}(G) ≤ 11/14 n for graphs with no isolated vertices or edges.

Improved the upper bound from 4/5 n to 11/14 n for the game total domination number.

Confirmed the conjecture that the upper bound could be less than 3/4 n in certain graph classes.

Abstract

The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph $G$. Each vertex chosen must strictly increase the number of vertices totally dominated. This process eventually produces a total dominating set of $G$. Dominator wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game total domination number of $G$, $\gamma_{{\rm tg}}(G)$, is the number of vertices chosen when Dominator starts the game and both players play optimally. Recently, Henning, Klav\v{z}ar, and Rall proved that $\gamma_{{\rm tg}}(G) \le \frac{4}{5}n$ holds for every graph $G$ which is given on $n$ vertices such that every component of it is of order at least $3$; they also conjectured that the sharp upper bound would be $\frac{3}{4}n$. Here, we prove that $\gamma_{{\rm tg}}(G)\le \frac{11}{14}n$ holds for every $G$ which contains no isolated vertices or isolated edges.