How does the game domination number behave by atomic changes on the graph?

Paul Dorbec
LaBRI, Univ. Bordeaux - CNRS



Content: In the domination game, two players alternate turns marking a vertex in a graph. When it is marked, a vertex must dominate at least one new vertex, either itself or one of its neighbours. When the set of marked vertices forms a dominating set (not necessarily minimal), the game comes to an end and the score of the game is the order of this set. One player, Dominator, tries to minimize this score while the other player, Staller, tries to maximize it. The domination game number $\gamma_g(G)$ is the score when both players play optimally and Dominator starts, $\gamma'_g(G)$ denoting the score when Staller starts. During this talk, we survey some recent results on the domination game, focusing on how the domination game number changes when the graph is subject to small changes such as deletion, addition or contraction of edges/vertices. We in particular show that these parameters show heredity only for edge contraction, but that bounds can be proved in most other cases. The results presented come from joint works with B. Bre\v sar, T. James, S. Klav\v zar, G. Ko\v smrlj, G. Renault, and A.~Vijayakumar.

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