Strong equality of Roman and perfect Roman domination in trees

A Roman dominating function (RD-function) on a graph $G = (V, E)$ is a function $f: V \longrightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. An Roman dominating function $f$ in a graph $G$ is perfect Roman dominating function (PRD-function) if every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The (perfect) Roman domination number $\gamma_R(G)$ ($\gamma_{R}^{p}(G)$) is the minimum weight of an (perfect) Roman dominating function on $G$. We say that $\gamma_{R}^{p}(G)$ strongly equals $\gamma_R(G)$, denoted by $\gamma_{R}^{p}(G)\equiv \gamma_R(G)$, if every RD-function on $G$ of minimum weight is a PRD-function. In this paper we show that for a given graph $G$, it is NP-hard to decide whether $\gamma_{R}^{p}(G)= \gamma_R(G)$ and also we provide a constructive characterization of trees $T$ with $\gamma_{R}^{p}(T)\equiv \gamma_R(T)$.

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.

New Bounds on the Triple Roman Domination Number of Graphs

In this paper, we derive sharp upper and lower bounds on the sum γ 3 R G + γ 3 R G ¯ and product γ 3 R G γ 3 R G ¯ , where G ¯ is the complement of graph G . We also show that for each tree T of order n ≥ 2 , γ 3 R T ≤ 3 n + s T / 2 and γ 3 R T ≥ ⌈ 4 n T + 2 − ℓ T / 3 ⌉ , where s T and ℓ T are the number of support vertices and leaves of T .

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).

Resolving Roman domination in graphs

Let [Formula: see text] be a graph and [Formula: see text] be a Roman dominating function defined on [Formula: see text]. Let [Formula: see text] be some ordering of the vertices of [Formula: see text]. For any [Formula: see text], [Formula: see text] is defined by [Formula: see text]. If for all [Formula: see text], [Formula: see text], we have [Formula: see text], that is [Formula: see text], for some [Formula: see text], then [Formula: see text] is called a resolving Roman dominating function (RDF) on [Formula: see text]. The weight of a resolving RDF [Formula: see text] on [Formula: see text] is [Formula: see text]. The minimum weight of a resolving RDF on [Formula: see text] is called the resolving Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. A resolving RDF on [Formula: see text] with weight [Formula: see text] is called a [Formula: see text]-function on [Formula: see text]. In this paper, we find the resolving Roman domination number of certain well-known classes of graphs. We also categorize the class of graphs whose resolving Roman domination number equals their order.

Graphs whose weak Roman domination number increases by the deletion of any edge

Let [Formula: see text] be a function on a graph [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is said to be undefended with respect to [Formula: see text] if it is not adjacent to a vertex [Formula: see text] with [Formula: see text]. A function [Formula: see text] is called a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text], such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text], has no undefended vertex. The weight of a WRDF is the sum of its function values over all vertices, and the weak Roman domination number [Formula: see text] is the minimum weight of a WRDF in [Formula: see text]. In this paper, we consider the effects of edge deletion on the weak Roman domination number of a graph. We show that the deletion of an edge of [Formula: see text] can increase the weak Roman domination number by at most 1. Then we give a necessary condition for [Formula: see text]-ER-critical graphs, that is, graphs [Formula: see text] whose weak Roman domination number increases by the deletion of any edge. Restricted to the class of trees, we provide a constructive characterization of all [Formula: see text]-ER-critical trees.

On the Quasi-Total Roman Domination Number of Graphs

Domination theory is a well-established topic in graph theory, as well as one of the most active research areas. Interest in this area is partly explained by its diversity of applications to real-world problems, such as facility location problems, computer and social networks, monitoring communication, coding theory, and algorithm design, among others. In the last two decades, the functions defined on graphs have attracted the attention of several researchers. The Roman-dominating functions and their variants are one of the main attractions. This paper is a contribution to the Roman domination theory in graphs. In particular, we provide some interesting properties and relationships between one of its variants: the quasi-total Roman domination in graphs.

Roman Domination of Some Chemical Graphs

The concept of Domination in graphs has application to the study of DNA structures. For investigating the chemical and physical properties, several topological indices used are Wiener index, Randic index, Zagreb index, Kier & Hall index that depends on vertex degree and distance sum, and have been used extensively for QSAR and QSPR studies. A Roman Dominating Function of G is function f: V→ {0, 1, 2} such that every vertex v for which f (v) = 0 has a neighbor u with f(u) = 2. The weight of a Roman dominating function f is w (f) = . The Roman domination number of a graph G is denoted by (G) and is the minimum weight of all possible Roman dominating functions. In this paper, we find Roman domination number of some chemicals graphs such as saturated hydrocarbons and unsaturated hydrocarbons, hexagonal chain, pyrene, Hexabenzocoronene, H-Phenylenic nanotube and N-Napthelenic nanotube.

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.