Roman κ-domination revisited

Kammerling and Volkmann [J. Korean Math. Soc. 46 (2009), 1309–1318] introduced the concept of Roman k-domination in graphs. For a fixed positive integer k, a function f: V (G) → {0,1,2} is a Roman k-dominating function on G if every vertex valued 0 under f is adjacent to at least k vertices valued 2 under f. In this paper, inspired by the concept of alliances in graphs, we revisit the concept of Roman k-domination by not-fixing k. 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.