Double Roman Graphs in P(3k, k)

Abstract

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P(3k,k), and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P(3k,k). This implies that P(3k,k) is a double Roman graph if and only if either k≡0 (mod 3) or k∈{1,4}.

On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)

“…Double Roman domination of families P(ck, k) has been studied recently for small k, including c = 3, 4, and 5. Shao et al [15] considered the double Roman domination number in generalized Petersen graphs P(3k, k) and constructed solutions providing the upper bounds, which gives exact values for γ dR (P(3k, k)).…”

“…In particular, generalized Petersen graphs and certain subfamilies of generalized Petersen graphs have been studied extensively in recent years. The results listed in subsection on related previous work include closed expressions for the double Roman domination number of some, and tight bounds for other subfamilies [12][13][14][15]. For more results on double Roman domination we refer to recent papers [16][17][18][19] and the references there.…”

On 3-Rainbow Domination Number of Generalized Petersen Graphs P(6k,k)

“…Generalized Petersen graphs have been studied extensively in the past, often as very interesting examples in research of various graph invariants. The interest seems to be even more intensive recently, the problems studied including domination [19], independent rainbow domination [20,21], Italian domination [22], Roman and double Roman domination [12,23,24], to name just a few. Many papers focus on subfamilies of Petersen graphs.…”

“…Optimal 3RD functions (a) of weight 15 for P(12, 2), (b) of weight 27 for P(24,4), and (c) of weight 39 for P(36, 6).…”