On domination game stable graphs and domination game edge-critical graphs

Xu, Kexiang; Li, Xia

doi:10.1016/j.dam.2018.05.027

Cited by 26 publications

(15 citation statements)

References 22 publications

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“…One of the widely studied problems regarding this game is the 3/5 Conjecture, posed and studied in [19], and later in [5,6,7,13,16]. The other directions of investigations are, among others, graphs critical with respect to this game ( [8,26]), realisations of the game domination numbers ( [22]) and computational complexity of such a game ( [20]). Further results on this topic can be found in [2,3,4,10,21,23,27].…”

Section: Introductionmentioning

confidence: 99%

Connected domination game

Borowiecki

Fiedorowicz

Sidorowicz

2019

Appl Anal Discrete Math

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In this paper we introduce a domination game based on the notion of connected domination. Let G = (V, E) be a connected graph of order at least 2. We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller. The players alternate taking turns choosing a vertex of G (Dominator starts). A move of a player by choosing a vertex v is legal, if (1) the vertex v dominates at least one additional vertex that was not dominated by the set of previously chosen vertices and (2) the set of all chosen vertices induces a connected subgraph of G. The game ends when none of the players has a legal move (i.e., G is dominated). The aim of Dominator is to finish as soon as possible, Staller has an opposite aim. Let D be the set of played vertices obtained at the end of the connected domination game (D is a connected dominating set of G). The connected game domination number of G, denoted γcg(G), is the minimum cardinality of D, when both players played optimally on G. We provide an upper bound on γcg(G) in terms of the connected domination number. We also give a tight upper bound on this parameter for the class of 2-trees. Next, we investigate the Cartesian product of a complete graph and a tree, and we give exact values of the connected game domination number for such a product, when the tree is a path or a star. We also consider some variants of the game, in particular, a Staller-start game.

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Section: Introductionmentioning

confidence: 99%

Connected domination game

Borowiecki

Fiedorowicz

Sidorowicz

2019

Appl Anal Discrete Math

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“…All the bounds in ( 1) and ( 2) are sharp [7,18]. For more information on the game (total) domination number of graphs with one vertex predominated see [11,15,16,24].…”

Section: Introductionmentioning

confidence: 99%

Predominating a vertex in the connected domination game

Bujtás¹,

Iršič²,

Klavžar³

2021

Preprint

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The connected domination game is played just as the domination game, with an additional requirement that at each stage of the game the vertices played induce a connected subgraph. The number of moves in a D-game (an S-game, resp.) on a graph G when both players play optimally is denoted by γ cg (G) (γ ′ cg (G), resp.). Connected Game Continuation Principle is established as a substitute for the classical Continuation Principle which does not hold for the connected domination game. Let G|x denote the graph G together with a declaration that the vertex x is already dominated. The first main result asserts that if G is a graph with γ cg (G) ≥ 3 and x ∈ V (G), then γ cg (G|x) ≤ 2γ cg (G) − 3 and the bound is sharp. The second main theorem states that if G is a graph with n(G) ≥ 2 and x ∈ V (G), then γ cg (G|x) ≥ 1 2 γ cg (G) and the bound is sharp. Graphs G and its vertices x for which γ ′ cg (G|x) = ∞ holds are also characterized.

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“…Following [5], the domination game has been studied further in many papers, see e.g. [3,4,6,7,15,18,24,25,26,27,28,29,30]. The notion also inspired the introduction of the total domination game on graphs [8,14,19,20,21,22,23], transversal game [9,10], disjoint domination game [12], connected domination game [2], and domination game on hypergraphs [11].…”

Section: Introductionmentioning

confidence: 99%

Fractional Domination Game

Bujtás¹,

Tuza²

2019

Electron. J. Combin.

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Given a graph $G$, a real-valued function $f: V(G) \rightarrow [0,1]$ is a fractional dominating function if $\sum_{u \in N[v]} f(u) \ge 1$ holds for every vertex $v$ and its closed neighborhood $N[v]$ in $G$. The aim is to minimize the sum $\sum_{v \in V(G)} f(v)$. A different approach to graph domination is the domination game, introduced by Brešar et al. [SIAM J. Discrete Math. 24 (2010) 979–991]. It is played on a graph $G$ by two players, namely Dominator and Staller, who take turns choosing a vertex such that at least one previously undominated vertex becomes dominated. The game is over when all vertices are dominated. Dominator wants to finish the game as soon as possible, while Staller wants to delay the end. Assuming that both players play optimally and Dominator starts, the length of the game on $G$ is uniquely determined and is called the game domination number of $G$. We introduce and study the fractional version of the domination game, where the moves are ruled by the condition of fractional domination. Here we prove a fundamental property of this new game, namely the fractional version of the so-called Continuation Principle. Moreover, we present lower and upper bounds on the fractional game domination number of paths and cycles. These estimates are tight apart from a small additive constant. We also prove that the game domination number cannot be bounded above by any linear function of the fractional game domination number.

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On domination game stable graphs and domination game edge-critical graphs

Cited by 26 publications

References 22 publications

Connected domination game

Connected domination game

Predominating a vertex in the connected domination game

Fractional Domination Game

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