On the double Roman bondage numbers of graphs

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text] for some vertex [Formula: see text], then [Formula: see text] has at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text], and if [Formula: see text] then [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text]. The weight of a DRDF [Formula: see text] is the sum [Formula: see text]. The minimum weight of a DRDF on a graph [Formula: see text] is the double Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. The double Roman bondage number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all edge subsets [Formula: see text] such that [Formula: see text]. In this paper, we study the double Roman bondage number in graphs. We determine the double Roman bondage number in several families of graphs, and present several bounds for the double Roman bondage number. We also study the complexity issue of the double Roman bondage number and prove that the decision problem for the double Roman bondage number is NP-hard even when restricted to bipartite graphs.

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

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 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 equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).

Double Roman domination subdivision number in graphs

For a graph [Formula: see text], a double Roman dominating function is a function [Formula: see text] having the property that if [Formula: see text], then vertex [Formula: see text] must have at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor with [Formula: see text], and if [Formula: see text], then vertex [Formula: see text] must have at least one neighbor with [Formula: see text]. The weight of a double Roman dominating function [Formula: see text] is the value [Formula: see text]. The double Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], equals the minimum weight of a double Roman dominating function on [Formula: see text]. The double Roman domination subdivision number [Formula: see text] of a graph [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the double Roman domination number. In this paper, we first show that the decision problem associated with sd[Formula: see text] is NP-hard and then establish upper bounds on the double Roman domination subdivision number for arbitrary graphs.

Total double Roman domination numbers in digraphs

Let [Formula: see text] be a finite and simple digraph with vertex set [Formula: see text]. A double Roman dominating function (DRDF) on digraph [Formula: see text] is a function [Formula: see text] such that every vertex with label 0 has an in-neighbor with label 3 or two in-neighbors with label 2 and every vertex with label 1 have at least one in-neighbor with label at least 2. The weight of a DRDF [Formula: see text] is the value [Formula: see text]. A DRDF [Formula: see text] on [Formula: see text] with no isolated vertex is called a total double Roman dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertex. In this paper, we initiate the study of the total double Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some bounds for the total double Roman domination number and we determine the total double Roman domination number of some classes of digraphs.

Double Roman Graphs in P(3k, k)

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}.