Upper distance-two domination

Abstract. Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). A set \(S \subset V\) is a \(2\)-packing in \(G\) if for any two vertices \(u,v \in S\), the distance between them satisfies \(d(u,v) \gt 2\). The upper \(2\)-packing number \(P_2(G)\) is the maximum cardinality of a \(2\)-packing in \(G\). A set \(S \subset V\) is a dominating set for \(G\) if every vertex in \(V - S\) is adjacent to at least one vertex in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set in \(G\). A set \(S \subset V\) is a distance-\(2\) dominating set if for every vertex \(v \in V - S\) there exists a vertex \(u \in S\) such that \(d(u,v) \leq 2\). The upper distance-\(2\) domination number \(\Gamma_{\leq 2}(G)\) is the maximum cardinality of a minimal distance-\(2\) dominating set in \(G\). In this paper we establish two families of graphs \(G\) for which \(P_2(G) = \gamma(G) = \Gamma_{\leq 2}(G)\), which extend several well-known equalities of the form \(P_2(G) = \gamma(G)\).