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Domination analysis
About: Domination analysis is a research topic. Over the lifetime, 3219 publications have been published within this topic receiving 35833 citations.
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TL;DR: The purpose of this paper is to characterize these graphs, which are called domination balanced if its vertex set can be partitioned into β(G) subsets so that each subset is a smallest dominating set of the complement G of G.
Abstract: A set D of vertices in a graph is said to be a dominating set if every vertex not in D is adjacent to some vertex in D. The domination number β(G) of a graph G is the size of a smallest dominating set. G is called domination balanced if its vertex set can be partitioned into β(G) subsets so that each subset is a smallest dominating set of the complement G of G. The purpose of this paper is to characterize these graphs.
136 citations
TL;DR: Several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs.
Abstract: Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012
136 citations
TL;DR: An upper bound on the double Roman domination number of a connected graph G in terms of the order of G is presented and the graphs attaining this bound are characterized.
131 citations
TL;DR: Examples and properties of vertex domination-critical graphs are given, a method of constructing them is presented, and some open questions are posed.
Abstract: A dominating set in a graph G is a set of vertices D such that every vertex of G is either in D or is adjacent some vertex of D. The domination number Γ(G) of G is the minimum cardinality of any dominating set. A graph is vertex domination-critical if the removal of any vertex decreases its domination number. This paper gives examples and properties of vertex domination-critical graphs, presents a method of constructing them, and poses some open questions. In the process several results for arbitrary graphs are presented.
129 citations
TL;DR: It is proved that $\gamma_g(G) \le 7n/11$ when $G$ is an isolate-free $n$-vertex forest and that $G) is a forest of nontrivial caterpillars for any isolate-based graph.
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 $\gamma_g(G)$ when Dominator plays first and by $\gamma_g^\prime(G)$ when Staller plays first. We prove that $\gamma_g(G) \le 7n/11$ when $G$ is an isolate-free $n$-vertex forest and that $\gamma_g(G) \le \left\lceil7n/10\right\rceil$ for any isolate-free $n$-vertex graph. In both cases we conjecture that $\gamma_g(G) \le 3n/5$ and prove it when $G$ is a forest of nontrivial caterpillars. We also resolve conjectures of Bresar, Klavžar, and Rall by showing that always $\gamma_g^\prime(G)\le\gamma_g(G)+1$, that for $k\ge2$ there a...
124 citations