Game domination numbers of a disjoint union of paths and cycles

Ruksasakchai, Watcharintorn; Onphaeng, Kritkhajohn; Worawannotai, Chalermpong

doi:10.2989/16073606.2018.1521880

Cited by 11 publications

(6 citation statements)

References 5 publications

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“…For the rest of the proof set n = n(G). If G is a disjoint union of cycles, then applying [26,Corollary 18] we deduce that γ g (G) ≤ ⌈ n 2 ⌉ ≤ ⌈ 11 20 n⌉. From now on, we assume that G satisfies the conditions of the theorem and it is not a disjoint union of cycles.…”

Section: Traceable Line Graphsmentioning

confidence: 99%

“…The second paper delivered the Continuation Principle which is a tool used to derive many important results, and the 3/5-conjecture which asserts that if G is an isolate-free graph, then γ g (G) ≤ 3 5 n(G), where n(G) denotes the order of G. These two papers caused the area to flourish-well above fifty papers have already been written. We emphasize papers [12,21,23,24,26,28] and investigations of different variants of the game [2,5,8,13,15,17,20].…”

Section: Introductionmentioning

confidence: 99%

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$1/2$-conjectures on the domination game and claw-free graphs

Bujtás¹,

Iršič²,

Klavžar³

2020

Preprint

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Let γ g (G) be the game domination number of a graph G. Rall conjectured that if G is a traceable graph, then γ g (G) ≤ 1 2 n(G) . Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if δ(G) ≥ 2, then γ g (G) ≤ 1 2 n(G) . We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound γ g (G) ≤ 11 20 n(G) over the class of claw-free graphs of minimum degree at least 2. Computer experiments supporting the new conjecture and sharpness examples are also presented.

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Section: Traceable Line Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

$1/2$-conjectures on the domination game and claw-free graphs

Bujtás¹,

Iršič²,

Klavžar³

2020

Preprint

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“…In addition, there are researchers studying the domination game numbers on several classes of graphs such as paths, cycles, forests, disjoint union of graphs, split graphs, etc. (See References [4,[8][9][10][11] for examples).…”

Section: Introductionmentioning

confidence: 99%

Competition-Independence Game and Domination Game

Worawannotai

Ruksasakchai

2020

Mathematics

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The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as many turns as possible. The game ends when all vertices are dominated. The game domination number, denoted by γ g ( G ) (respectively γ g ′ ( G ) ), is the total number of turns when both players play optimally and when Dominator (respectively Staller) starts the game. In this paper, we study a version of this game where the set of chosen vertices is always independent. This version turns out to be another game known as the competition-independence game. The competition-independence game is played on a graph by two players, Diminisher and Sweller. They take turns in constructing maximal independent set M, where Diminisher tries to minimize | M | and Sweller tries to maximize | M | . Note that, actually, it is the domination game in which the set of played vertices is independent. The competition-independence number, denoted by I d ( G ) (respectively I s ( G ) ) is the optimal size of the final independent set in the competition-independence game if Diminisher (respectively Sweller) starts the game. In this paper, we check whether some well-known results in the domination game hold for the competition-independence game. We compare the competition-independence numbers to the game domination numbers. Moreover, we provide a family of graphs such that many parameters are equal. Finally, we present a realization result on the competition-independence numbers.

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“…Staller-start game domination number γ ′ g (G)) of G. The domination game was introduced in [6]. Early influential references on this game include [5,7,8,11,17,19], while from the very extensive recent development on the domination game and its variants we select the papers [2,3,9,14,16,21,22]. In this paper we are interested in the following conjecture that has been proposed several years ago by D. Rall, the first published source of it is [15,Conjecture 1.1].…”

Section: Introductionmentioning

confidence: 99%