The domination game played on unions of graphs

The total domination game is played on a graph [Formula: see text] by two players, named Dominator and Staller. They alternately select vertices of [Formula: see text]; each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. In this paper, we show that for any graph [Formula: see text] and a vertex [Formula: see text], where [Formula: see text] has no isolated vertex, we have [Formula: see text] and [Formula: see text]. Moreover, all such differences can be realized by some connected graphs.

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Progress towards the total domination game 34-conjecture

Discrete Mathematics ◽

10.1016/j.disc.2016.05.014 ◽

2016 ◽

Vol 339 (11) ◽

pp. 2620-2627 ◽

Cited By ~ 14

Author(s):

Michael A. Henning ◽

Douglas F. Rall

Keyword(s):

Total Domination ◽

Total Domination Game ◽

Domination Game

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Domination game on paths and cycles

Ars Mathematica Contemporanea ◽

10.26493/1855-3974.891.e93 ◽

2017 ◽

Vol 13 (1) ◽

pp. 125-136 ◽

Cited By ~ 14

Author(s):

Gašper Košmrlj

Keyword(s):

Domination Game

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Domination game critical graphs

Discussiones Mathematicae Graph Theory ◽

10.7151/dmgt.1839 ◽

2015 ◽

Vol 35 (4) ◽

pp. 781 ◽

Cited By ~ 13

Author(s):

Csilla Bujtás ◽

Sandi Klavžar ◽

Gašper Košmrlj

Keyword(s):

Critical Graphs ◽

Domination Game

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Guarded subgraphs and the domination game

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2123 ◽

2015 ◽

Vol Vol. 17 no. 1 (Graph Theory) ◽

Author(s):

Boštjan Brešar ◽

Sandi Klavžar ◽

Gasper Košmrlj ◽

Doug F. Rall

Keyword(s):

Graph Theory ◽

Domination Number ◽

Metric Properties ◽

International Audience ◽

Game Domination Number ◽

Domination Game

Graph Theory International audience 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.

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On graphs with small game domination number

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm160207003k ◽

2016 ◽

Vol 10 (1) ◽

pp. 30-45 ◽

Cited By ~ 4

Author(s):

Sandi Klavzar ◽

Gasper Kosmrlj ◽

Simon Schmidt

Keyword(s):

Domination Number ◽

Small Game ◽

Game Domination Number ◽

Domination Game

The domination game is played on a graph G by Dominator and Staller. The game domination number ?(G) of G is the number of moves played when Dominator starts and both players play optimally. Similarly, ?g (G) is the number of moves played when Staller starts. Graphs G with ?(G) = 2, graphs with ?g(G) = 2, as well as graphs extremal with respect to the diameter among these graphs are characterized. In particular, ?g (G) = 2 and diam(G) = 3 hold for a graph G if and only if G is a so-called gamburger. Graphs G with ?(G) = 3 and diam(G) = 6, as well as graphs G with ?g(G) = 3 and diam(G) = 5 are also characterized.

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