Extremal Problems for Game Domination Number

Abstract

In the domination game on a graph G, two players called Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of G. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the size of the dominating set produced is the game domination number of G, denoted by γ g (G) when Dominator plays first and by γ ′ g (G) when Staller plays first. We prove that γ g (G) ≤ 7n/11 when G is an isolate-free n-vertex forest and that γ g (G) ≤ ⌈7n/10⌉ for any isolate-free n-vertex graph. In both cases we conjecture that γ g (G) ≤ 3n/5 and prove it when G is a forest of nontrivial caterpillars. We also resolve conjectures of Brešar, Klavžar, and Rall by showing that always γ ′ g (G) ≤ γ g (G) + 1, that for k ≥ 2 there are graphs G satisfying γ g (G) = 2k and γ ′ g (G) = 2k − 1, and thatwhen G is a forest. Our results follow from fundamental lemmas about the domination game that simplify its analysis and may be useful in future research.

Characterisation of forests with trivial game domination numbers

“…It was shown in Kinnersley et al (2013), verifying a conjecture of Brešar et al (2010), that γ g (G) and γ g (G) can differ by at most one for any graph G. One might expect that γ g (G) is always at most γ g (G), but examples such as the 5-cycle C 5 show that this need not be true: one easily sees that γ (C 5 ) = 2 = γ g (C 5 ) but γ g (C 5 ) = 3. There are other examples as well.…”

“…There are other examples as well. For any cycle C n , there are explicit formulas for γ g (C n ) and γ g (C n ), stated in Košmrlj (2014) and attributed to the unpublished manuscript (Kinnersley et al 2012). Using these, one gets that γ g (C n ) < γ g (C n ) exactly when n ≥ 5 is congruent to 1 or 2 modulo 4.…”

“…A is a dominating set of G if it dominates G. The domination number γ (G) of G is the minimum size of a dominating set. There are numerous papers about this well known graph parameter; we restrict our attention to the so called 'Domination Game', introduced recently in Brešar et al (2010) and since elaborated on in Brešar et al (2014), Brešar et al (2015), Brešar et al (2013), Dorbec et al (2015), Henning et al (2014), Kinnersley et al (2013), and Košmrlj (2014).…”

On graphs with small game domination number

“…We next collect some known results (or part of the folklore results) to be used later on. A fundamental result about the domination game is the following theorem for which the fact that γ g (G) ≤ γ g (G) + 1 holds was proved in [4], while the inequality γ g (G) ≤ γ g (G) + 1 was later established in [15].…”

“…One of the reasons for this interest is the so-called 3/5-conjecture from [15] asserting that γ g (G) ≤ 3n/5 holds for any isolate free graph of order n. Trees that attain this bound were investigated in [5], while recently Bujtas [7] made a breakthrough by proving that the conjecture holds for all graphs with the minimum degree at least 3. In order to achieve this result she further developed the proof technique introduced in [6] where the conjecture is verified for all trees in which no two leaves are at distance 4.…”