Global Domination and Packing Numbers

Abstract

For a graph \(G = (V, E)\), \(X \subseteq V\) is a global dominating set if \(X\) dominates both \(G\) and the complement graph \(\bar{G}\). A set \(X \subseteq V\) is a packing if its pairwise members are distance at least \(3\) apart. The minimum number of vertices in any global dominating set is \(\gamma_g(G)\), and the maximum number in any packing is \(\rho(G)\). We establish relationships between these and other graphical invariants, and characterize graphs for which \(\rho(G) = \rho(\bar{G})\). Except for the two self-complementary graphs on \(5\) vertices and when \(G\) or \(\bar{G}\) has isolated vertices, we show \(\gamma_g(G) \leq \lfloor n/2 \rfloor\), where \(n = |V|\).