On locating-dominating set of regular graphs (Q2052122)

Summary: Let \(G\) be a simple, connected, and finite graph. For every vertex \(v\in V(G)\), we denote by \(N_G(v)\) the set of neighbours of \(v\) in \(G\). The locating-dominating number of a graph \(G\) is defined as the minimum cardinality of \(W\subseteq V(G)\) such that every two distinct vertices \(u,v\in V(G)\backslash W\) satisfies \(\emptyset\neq N_G(u)\cap W\neq N_G(v)\cap W\neq\emptyset\). A graph \(G\) is called \(k\)-regular graph if every vertex of \(G\) is adjacent to \(k\) other vertices of \(G\). In this paper, we determine the locating-dominating number of \(k\)-regular graph of order \(n\), where \(k=n-2\) or \(k=n-3\).