## Game Total Domination and Total Dominating Sequences

- April 1, 2016
- 2:30 p.m.
- LeConte 312

## Abstract

Two players, Dominator and Staller, alternate choosing a vertex from a graph *G* that has no isolated vertices. Each vertex chosen by a player must totally dominate (i.e., must be adjacent to ) at least one vertex that was not totally dominated by the set of vertices chosen previously in the game. The game ends when no such vertex can be chosen. Both players use an optimal strategy--Dominator to end the game as quickly as possible and Staller to prolong the game. If Dominator is the first to choose a vertex, the number of vertices chosen is the **game total domination number** of *G*, denoted $\gamma _ {tg}(G)$. I will present some of the basic results about the game total domination number as well as some progress on proving the conjectured upper bound of 3*n*/4 on this invariant for a graph of order *n*.

If both players are attempting to make the game on *G* last as long as possible, the number of vertices chosen is called the **Grundy total domination number** of *G*. This is equivalent to the worst outcome from an *online* version of total domination in which vertices are processed one at a time and the only information available when a vertex is processed is its open neighborhood. I will present some results on lower and upper bounds for this graphical invariant for various classes of graphs.

This is joint work with various subsets of Boštjan Brešar, Mike Henning and Sandi Klavžar.