Abstract
The game total domination number, γtg, 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 γtg(G|v)≥γtg(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 γtg(G)≤γtg(G−v)+4 and γtg′(G)≤γtg′(G−v)+4.
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