Domination Game and an Imagination Strategy

Abstract

The domination game played on a graph G consists of two players, Dominator and Staller who alternate taking turns choosing a vertex from G such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible and Staller wishes to delay the process as much as possible. The game domination number γ g (G) is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number γ g (G) when Staller starts the game. An imagination strategy is developed as a general tool for proving results on the domination game. We show that for any graph G, γ(G) ≤ γ g (G) ≤ 2γ(G) − 1, and that all possible values can be realized. It is proved that for any graph G, γ g (G) − 1 ≤ γ g (G) ≤ γ g (G) + 2, and that most of the possibilities for mutual values of γ g (G) and γ g (G) can be realized. A connection with Vizing's conjecture is established, and a lower bound on the * The paper was initiated during the visit of the third author to the IMFM and completed during the visit of the first two authors at Furman University supported in part by the Slovenia-USA bilateral grant BI-US/08-10-004 and the Wylie Enrichment Fund of Furman University.† Supported by the Ministry of Science of Slovenia under the grants P1-0297 and J1-2043. The author is also with the Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana.game domination number of an arbitrary Cartesian product is proved. Several problems and conjectures are also stated.

Extremal Problems for Game Domination Number

“…Brešar, Klavžar, and Rall [2] introduced a game variant of graph domination, which they attributed to Henning. In the domination game on a graph G, two players called Dominator and Staller take turns choosing vertices of G. Each added vertex must dominate at least one vertex not dominated by previously chosen vertices.…”

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

“…Edge-hitter's first r −1 moves in V (C ) all decrease the weight by 4, while his rth move in V (C ) decreases the weight by 5. Thus, 4 , and Edge-hitter can play according to rule (R5), then he can achieve his Proof. Suppose that Edge-hitter can play according to (R5).…”

“…[15]) and independently by Bodlaender [2] for general graphs; it has seen extensive study, see the survey [28]. Recently, work has been done on competitive optimization variants of list-colouring [5] and its more studied related version called paintability as introduced in [26] (for further references see Section 8 of [28]), matching [11], domination [4], total domination [19], disjoint domination [8], Ramsey theory [9,16,17], and more [3].…”

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].…”

“…The game was introduced in [4] and already received a considerable attention. 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.…”