### Guarded subgraphs and the domination game

*Boštjan Brešar, Sandi Klavžar, Gasper Košmrlj, Doug F. Rall*

#### Abstract

We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric
subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are
obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller,
alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated
so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is
just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game
and both players play optimally. The main result of this paper is that the game domination number of a graph is not
smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.

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