The equidistant dimension of graphs: NP-completeness and the case of lexicographic product graphs

• Published in: AIMS mathematics Vol. 9; no. 6; pp. 15325 - 15345
• Main Authors: Gispert-Fernández, Adrià, Rodríguez-Velázquez, Juan Alberto
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2024
• Subjects: distance-equalizer; distances in graphs; equidistant dimension; lexicographic product; np-complete problem
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2024744

Let $ V(G) $ be the vertex set of a simple and connected graph $ G $. A subset $ S\subseteq V(G) $ is a distance-equalizer set of $ G $ if, for every pair of vertices $ u, v\in V(G)\setminus S $, there exists a vertex in $ S $ that is equidistant to $ u $ and $ v $. The minimum cardinality among the distance-equalizer sets of $ G $ is the equidistant dimension of $ G $, denoted by $ \xi(G) $. In this paper, we studied the problem of finding $ \xi(G\circ H) $, where $ G\circ H $ denotes the lexicographic product of two graphs $ G $ and $ H $. The aim was to express $ \xi(G\circ H) $ in terms of parameters of $ G $ and $ H $. In particular, we considered the cases in which $ G $ has a domination number equal to one, as well as the cases where $ G $ is a path or a cycle, among others. Furthermore, we showed that $ \xi(G)\le \xi(G\circ H)\le \xi(G)|V(H)| $ for every connected graph $ G $ and every graph $ H $ and we discussed the extreme cases. We also showed that the general problem of finding the equidistant dimension of a graph is NP-hard.