Roman κ-domination revisited

Abstract

Kammerling and Volkmann [J. Korean Math. Soc. 46 (2009) 1309–1318] introduced the concept of Roman κ-domination in graphs. For a fixed positive integer κ, a function f : V(G) → {0, 1, 2} is a Roman κ-dominating function on G if every vertex valued 0 under f is adjacent to at least κ vertices valued 2 under f. In this paper, inspired by the concept of alliances in graphs, we revisit the concept of Roman κ-domination by not-fixing κ. We prove upper bounds for the new variant in cactus graphs and characterize cactus graph achieving equality for the given bound. We also present a probabilistic upper bound for this variant.