The double Roman domatic number of a digraph

Abstract

A double Roman dominating function on a digraph D with vertex set V (D) is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function f : V (D) → {0, 1, 2, 3} having the property that if f (v) = 0, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor w with f (w) = 3, and if f (v) = 1, then the vertex v must have at least one in-neighbor u with f (u) ≥ 2. A set {f 1 , f 2 ,. .. , f d } of distinct double Roman dominating functions on D with the property that d i=1 f i (v) ≤ 3 for each v ∈ V (D) is called a double Roman dominating family (of functions) on D. The maximum number of functions in a double Roman dominating family on D is the double Roman domatic number of D, denoted by d dR (D). We initiate the study of the double Roman domatic number, and we present different sharp bounds on d dR (D). In addition, we determine the double Roman domatic number of some classes of digraphs.

Total Italian domatic number of graphs

“…, is called a total Italian dominating family (of functions) on G. The maximum number of functions in a total Italian dominating family on G is the total Italian domatic number d tI (G) of G. Italian domatic number has been studied in [12], [14].…”

Total Roman2-Reinforcement of Graphs

“…e R2DF was introduced in [2] and has been studied by several authors [3][4][5][6][7][8][9][10][11][12][13]. For more details on R2D-number, we refer the reader to the recent book chapter [14] and survey paper [15].…”