The distance Roman domatic number of a graph
Hamideh Aram, Seyed Mahmoud Sheikholeslami, Lutz Volkmann, S. Arumugam · AKCE International Journal of Graphs and Combinatorics · 2012
Let k be a positive integer, and let G be a simple graph with vertex set V(G). A k-distance Roman dominating function on G is a labeling f: V(G) → {0, 1, 2} such that every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, …, fdd} of k-distance Roman dominating functions on G with the property that eqfi(v) ≤ 2 for each v ∊ V(G), is called a k-distance Roman dominating family (of functions) on G. The maximum number of functions in a k-distance Roman dominating family on G is the k-distance Roman domatic number of G, denoted by . In this paper we initiate the study of k-distance Roman domatic number in graphs and we present some sharp bounds for . In addition, we determine the k-distance Roman domatic number of some graphs.