Locating-Domination in Complementary Prisms

Let \(G = (V, E)\) be a graph and \(\overline{G}\) be the complement of \(G\). The complementary prism of \(G\), denoted \(G \overline{G}\), is the graph formed from the disjoint union of \(G\) and \(\overline{G}\) by adding the edges of a perfect matching between the corresponding vertices of \(G\) and \(\overline{G}\). A set \(D \subseteq V(G)\) is a locating-dominating set of \(G\) if for every \(u \in V(G) \setminus D\), its neighborhood \(N(u) \cap D\) is nonempty and distinct from \(N(v) \cap D\) for all \(v \in V(G) \setminus D\) where \(v \neq u\). The locating-domination number of \(G\) is the minimum cardinality of a locating-dominating set of \(G\). In this paper, we study locating-domination of complementary prisms. We determine the locating-domination number of \(G \overline{G}\) for specific graphs \(G\) and characterize the complementary prisms with small locating-domination numbers. We also present upper and lower bounds on the locating-domination numbers of complementary prisms, and we show that all values between these bounds are achievable.