Effect of predomination and vertex removal on the game total domination number of a graph

TL;DR

This paper investigates how vertex predomination and removal influence the game total domination number in graphs, establishing bounds and introducing new variations of the total domination game.

Contribution

It proves bounds on the change in the game total domination number due to vertex predomination and removal, and introduces new game variations.

Findings

Vertex predomination can decrease the game total domination number by at most 2.

Removing a vertex increases the game total domination number by at most 4.

Infinite families of graphs attain the bounds for predomination effects.

Abstract

The game total domination number, ${\gamma_{g}^{t}}$, was introduced by Henning et al.\ in 2015. In this paper we study the effect of vertex predomination on the game total domination number. We prove that ${\gamma_{g}^{t}}(G|v) \geq {\gamma_{g}^{t}}(G) - 2$ holds for all vertices $v$ of a graph $G$ and present infinite families attaining the equality. To achieve this, some new variations of the total domination game are introduced. The effect of vertex removal is also studied. We show that ${\gamma_{g}^{t}}(G) \leq {\gamma_{g}^{t}}(G-v) + 4$ and ${\gamma_{g}^{t}}'(G) \leq {\gamma_{g}^{t}}'(G-v) + 4$.