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.

Source : oai:HAL:hal-01196867v1

Volume: Vol. 17 no. 1

Section: Graph Theory

Published on: March 27, 2015

Submitted on: October 18, 2013

Keywords: domination game,game domination number,convex subgraph,(2-)isometric subgraph,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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