Geodesic
Characterizations of minimal dominating sets and the well-dominated property in lexicographic product graphs
Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 1.

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A graph is said to be well-dominated if all its minimal dominating sets are of the same size. The class of well-dominated graphs forms a subclass of the well studied class of well-covered graphs. While the recognition problem for the class of well-covered graphs is known to be co-NP-complete, the recognition complexity of well-dominated graphs is open. In this paper we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and derive a characterization of the well-dominated lexicographic product graphs. As a side result motivated by this study, we give a polynomially testable characterization of well-dominated graphs with domination number two, and show, more generally, that well-dominated graphs can be recognized in polynomial time in any class of graphs with bounded domination number. Our results include a characterization of dominating sets in lexicographic product graphs, which generalizes the expression for the domination number of such graphs following from works of Zhang et al. (2011) and of \v{S}umenjak et al. (2012). 
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     title = {Characterizations of minimal dominating sets and the well-dominated property in lexicographic product graphs},
     journal = {Discrete mathematics & theoretical computer science},
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Gözüpek, Didem; Hujdurović, Ademir; Milanič, Martin. Characterizations of minimal dominating sets and the well-dominated property in lexicographic product graphs. Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 1. doi : 10.23638/DMTCS-19-1-25. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-19-1-25/

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