Progress Towards the Total Domination Game $\frac{3}{4}$-Conjecture

TL;DR

This paper advances the understanding of the total domination game in graphs by proving the 3/4-Conjecture for a specific class of graphs, showing that Dominator can efficiently complete the game.

Contribution

It proves the 3/4-Conjecture for graphs with certain degree sum and distance conditions, extending previous results in total domination game theory.

Findings

Dominator can complete the game in at most 3n/4 moves under specified conditions.

The conjecture holds for graphs with minimum degree at least 2 and additional structural constraints.

The paper introduces a greedy strategy that guarantees the bound.

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, $\gamma_{\rm tg}(G)$, of $G$ is the number of vertices chosen when Dominator starts the game and both players play optimally. Henning, Klav\v{z}ar and Rall [Combinatorica, to appear] posted the $\frac{3}{4}$-Game Total Domination Conjecture that states that if $G$ is a graph on $n$ vertices in which every component contains at least three vertices, then $\gamma_{\rm tg}(G) \le \frac{3}{4}n$. In this paper, we prove this conjecture over the class of graphs $G$ that satisfy both the condition that the degree sum of adjacent vertices in $G$ is at least $4$ and the condition that no two vertices of degree $1$ are at distance $4$ apart in $G$. In particular, we prove that by adopting a greedy strategy, Dominator can complete the total domination game played in a graph with minimum degree at least $2$ in at most $3n/4$ moves.