Roman Dominating Function Research Articles

Convex Roman Dominating Functions on Graphs under some Binary Operations

Let $G$ be a connected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by $\omega_{G}^{CvR}(f)$, is given by $\omega_{G}^{CvR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by $\gamma_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we specifically study the concept of convex Roman domination in the corona and edge corona of graphs, complementary prism, lexicographicproduct, and Cartesian product of graphs.

The Novel Concept of Roman Domination in Fuzzy Graph

Suppose = , is a graph. A Roman dominating function (RDF) of is a function: → {0,1,2} such that every vertex for which = 0 has a neighbor with = 2. The weight of an RDF is = Σ ∈ . The Roman domination number (RDN) of a graph , denoted by ℘, is the minimum weight of all possible RDFs. In this study, we define RDF on fuzzy graph (FG). Our purpose is to develop a notion of the RDF and also to present some basic definitions, notations, remarks, and proofs related to RDF on FGs.

Hardness results of global roman domination in graphs

A Roman dominating function (RDF) of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. A global Roman dominating function (GRDF) of a graph G=(V,E) is a function f:V→{0,1,2} such that f is a Roman dominating function of both G and its complement G¯. The weight of f is f(V)=Σu∈Vf(u). The minimum weight of a GRDF in a graph G is known as global Roman domination number of G and is denoted by γgR(G). Minimum Global Roman Domination is to find a global Roman dominating function of minimum weight and Decide Global Roman Domination is the decision version of Minimum Global Roman Domination. In this paper, we show that Decide Global Roman Domination is NP-complete for bipartite graphs and chordal graphs. On the positive side, we give a polynomial-time algorithm for Minimum Global Roman Domination for threshold graphs. Further, we propose an O(ln|V|)-approximation algorithm for Minimum Global Roman Domination for a graph G=(V,E). We also show that Minimum Global Roman Domination cannot be approximated within a factor of (1−ϵ)ln|V| for any ϵ>0 unless P = NP. Next, we show that Minimum Global Roman Domination admits a constant approximation algorithm for bounded degree graphs. Finally, we show that Minimum Global Roman Domination is APX-complete for bounded degree graphs.

Convex Roman Dominating Function in Graphs

Let $G$ be a connected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by $\omega_{G}^{CvR}(f)$, is given by $\omega_{G}^{CvR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by $\gamma_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we determine the convex Roman domination numbers of some graphs and give some realization results involving convex Roman domination, connected Roman domination, and convex domination numbers.

Algorithmic results in Roman dominating functions on graphs

Given a graph G=(V,E), a subset D⊆V (respectively, function f:V→{0,1,2}) is a dominating set (DS) (respectively, Roman dominating function (RDF)) of G if each vertex v∈V∖D (respectively, v∈V with f(v)=0) is adjacent to a vertex u∈D (respectively, u∈V with f(u)=2). The domination number of G is the minimum cardinality of an DS of G and the Roman domination number of G is the minimum weight of an RDF f of G, where the weight of f is ∑v∈Vf(v). The (Roman) domination problem is to compute the (Roman) domination number of a given graph. In this paper, we study the Roman domination problem. We show that the complexity of the problem differs from the complexity of the domination problem and the problem is NP-complete for circle graphs and undirected path graphs and is APX-complete for graphs of degree at most 4. We also propose an integer linear programming (ILP) formulation with polynomial number of constraints for the problem. Additionally, we use the ILP formulation to give an H(Δ(G)+1)-approximation algorithm for solving the problem for any graph G, where Δ(G) is the maximum degree of G. Furthermore, we show that the optimization version of the problem on split and chordal graphs cannot be approximated in polynomial time within (1/2−ε)ln⁡|V| for any ε>0, unless NP ⊆ DTIME (|V|O(log⁡log⁡|V|)).

On the Roman domination problem of some Johnson graphs

A Roman domination function (RDF) on a graph G with a set of vertices V = V(G) is a function f : V ? {0, 1, 2} which satisfies the condition that each vertex v ? V such that f (v) = 0 is adjacent to at least one vertex u such that f (u) = 2. The minimum weight value of an RDF on graph G is called the Roman domination number (RDN) of G and it is denoted by ?R(G). An RDF for which ?R(G) is achieved is called a ?R(G)-function. This paper considers Roman domination problem for Johnson graphs Jn,2 and Jn,3. For Jn,2, n ? 4 it is proved that ?R(Jn,2) = n ? 1. New lower and upper bounds for Jn,3, n ? 6 are derived using results on the minimal coverings of pairs by triples. These bounds quadratically depend on dimension n.

Restrained Roman and restrained Italian domatic numbers of graphs

Let G be a graph with vertex set V ( G ) . A Roman dominating function (RDF) on a graph G is a function f : V ( G ) ⟶ { 0 , 1 , 2 } satisfying the condition that each vertex u with f ( u ) = 0 has a neighbor v with f ( v ) = 2 . An Italian dominating function (IDF) is a function f : V ( G ) ⟶ { 0 , 1 , 2 } having the property that f ( N ( u ) ) ≥ 2 for every vertex u with f ( u ) = 0 , where N ( u ) is the neighborhood of u . If f is an RDF or an IDF on G , then let V 0 = { v ∈ V ( G ) : f ( v ) = 0 } . A restrained Roman (Italian) dominating function is an RDF (IDF) f having the property that the subgraph induced by V 0 does not have an isolated vertex. A set { f 1 , f 2 , … , f d } of distinct restrained Roman (Italian) dominating functions on G with the property that ∑ i = 1 d f i ( v ) ≤ 2 for each v ∈ V ( G ) is called a restrained Roman (Italian) dominating family (of functions) on G . The maximum number of functions in a restrained Roman (Italian) dominating family on G is the restrained Roman (Italian) domatic number of G , denoted by d r R ( G ) ( d r I ( G ) ). We initiate the study of the restrained Roman (Italian) domatic numbers, and we present different sharp bounds on d r R ( G ) and d r I ( G ) . In addition, we determine these parameters for some classes of 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.

ALGORITHMIC ASPECTS OF ROMAN GRAPHS

Let $G=(V, E)$ be a graph. A set $S subseteq V$ is called a dominating set of $G$ if for every $vin V-S$ there is at least one vertex $u in N(v)$ such that $uin S$. The domination number of $G$, denoted by $gamma(G)$, is equal to the minimum cardinality of a dominating set in $G$. A Roman dominating function (RDF) on $G$ is a function $f:Vlongrightarrow{0,1,2}$ such that every vertex $vin V$ with $f(v)=0$ is adjacent to at least one vertex $u$ with $f(u)=2$. The weight of $f$ is the sum $f(V)=sum_{vin V}f (v)$. The minimum weight of a RDF on $G$ is the Roman domination number of $G$, denoted by $gamma_R(G)$. A graph $G$ is a Roman Graph if $gamma_R(G)=2gamma(G)$. In this paper, we first study the complexity issue of the problem posed in [E.J. Cockayane, P.M. Dreyer Jr., S.M. Hedetniemi and S.T. Hedetniemi, On Roman domination in graphs, textit{Discrete Math.} 278 (2004), 11--22], and show that the problem of deciding whether a given graph is a Roman graph is NP-hard even when restricted to chordal graphs. Then, we give linear algorithms that compute the domination number and the Roman domination number of a given unicyclic graph. Finally, using these algorithms we give a linear algorithm that decides whether a given unicyclic graph is a Roman graph.

Weak Roman subdivision number of graphs

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex u for which [Formula: see text] is adjacent to at least one vertex v for which [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is undefended if it is not adjacent to a vertex with [Formula: see text]. The function [Formula: see text] is 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] if [Formula: see text] has no undefended vertex. The Roman domination subdivision number of a graph [Formula: see text] is the minimum number of edges that must be subdivided in order to increase the Roman domination number of [Formula: see text]. We introduce the concept of weak Roman subdivision number of a graph [Formula: see text], denoted by [Formula: see text] as the minimum number of edges that must be subdivided in order to increase the weak Roman domination number of [Formula: see text]. In this paper, we determine the exact values of the weak Roman subdivision number for paths, cycles and complete bipartite graphs. We obtain bounds for the weak Roman subdivision number of a graph and characterize the extremal graphs.

Powers of Large Matrices on GPU Platforms to Compute the Roman Domination Number of Cylindrical Graphs

The Roman domination in a graph $G$ is a variant of the classical domination, defined by means of a so-called Roman domination function $f\colon V(G)\to \{0,1,2\}$ such that if $f(v)=0$ then, the vertex $v$ is adjacent to at least one vertex $w$ with $f(w)=2$ . The weight $f(G)$ of a Roman dominating function of $G$ is the sum of the weights of all vertices of $G$ , that is, $f(G)=\sum _{u\in V(G)}f(u)$ . The Roman domination number $\gamma _{R}(G)$ is the minimum weight of a Roman dominating function of $G$ . In this paper we propose algorithms to compute this parameter involving the $(\min,+)$ powers of large matrices with high computational requirements and the GPU (Graphics Processing Unit) allows us to accelerate such operations. Specific routines have been developed to efficiently compute the $(\min,+)$ product on GPU architecture, taking advantage of its computational power. These algorithms allow us to compute the Roman domination number of cylindrical graphs $P_{m}\Box ~C_{n}$ i.e., the Cartesian product of a path and a cycle, in cases $m=7,8$ ,9 $n\geq 3$ and $m\geq $ 10 $, n\equiv 0\pmod 5$ . Moreover, we provide a lower bound for the remaining cases $m\geq $ 10 $, n\not \equiv 0\pmod 5$ .

Majority Roman domination in graphs

A Majority Roman Dominating Function (MRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) the sum of its function values over at least half the closed neighborhood is at least one and (ii) every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a MRDF is the sum of its function values over all vertices. The Majority Roman Domination Number of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text]. In this paper, we initiate the study of Majority Roman Domination in Graphs.

On the Roman domination subdivision number of a graph

A Roman dominating function (RDF) of a graph G is a labeling $$f:V(G)\longrightarrow \{0,1,2\}$$ such that every vertex with label 0 has a neighbor with label 2. The weight of an RDF is the sum of its functions values over all vertices, and the Roman domination number of G is the minimum weight of an RDF of G. The Roman domination subdivision number $$\mathrm {sd}_{\gamma _{R}}(G)$$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number of G. In this paper, we present a new upper bound on the Roman domination subdivision number by showing that for every connected graph G of order at least three, $$\begin{aligned} \mathrm {sd}_{\gamma _{R}}(G)\le 3+\min \{\deg _2(v)\mid v\in V\;\mathrm {and} \;d(v)\ge 2\}, \end{aligned}$$ where $$\deg _2(v)$$ is the number of vertices of G at distance 2 from vertex v. Moreover, we show that the decision problem associated with $$\mathrm {sd}_{\gamma _{R}}(G)$$ is NP-hard for bipartite graphs.

Critical graphs with Roman domination number four

A Roman domination function on a graph G is a function satisfying the condition that every vertex u for which r(u) = 0 is adjacent to at least one vertex v for which r(v) = 2. The weight of a Roman domination function is the value The Roman domination number of G is the minimum weight of a Roman domination function on G. “Roman Criticality” often refers to the study of graphs where the Roman domination number decreases when adding an edge or removing a vertex of the graph. In this paper we add some condition to this notion of criticality and give a complete characterization of critical graphs with Roman Domination number

Algorithmic aspects of Roman domination in graphs

For a simple, undirected graph $$G = (V, E)$$ , a Roman dominating function (RDF) $$f{:}V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that, every vertex u with $$f(u) = 0$$ is adjacent to at least one vertex v for which $$f(v) = 2$$ . The weight of a RDF is the sum $$f(V) = \sum _{v \in V}f(v)$$ . The minimum weight of a RDF is called the Roman domination number and is denoted by $$\gamma _{R}(G)$$ . Given a graph G and a positive integer k, the Roman domination problem (RDP) is to check whether G has a RDF of weight at most k. The RDP is known to be NP-complete for bipartite graphs. We strengthen this result by showing that this problem remains NP-complete for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. We show that $$\gamma _{R}(G)$$ is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. The minimum Roman domination problem (MRDP) is to find a RDF of minimum weight in the input graph. We show that the MRDP for star convex bipartite graphs and comb convex bipartite graphs cannot be approximated within $$(1 - \epsilon ) \ln |V|$$ for any $$\epsilon > 0$$ unless $$NP \subseteq DTIME(|V|^{O(\log \log |V|)})$$ and also propose a $$2(1+\ln (\varDelta +1))$$ -approximation algorithm for the MRDP, where $$\varDelta $$ is the maximum degree of G. Finally, we show that the MRDP is APX-complete for graphs with maximum degree 5.

Algorithmic and complexity aspects of problems related to total Roman domination for graphs

A function $$f:V(G)\rightarrow \{0,1,2\}$$ is a Roman dominating function (RDF) if every vertex u for which $$f(u)=0$$ is adjacent to at least one vertex v for which $$f(v)=2$$. The weight of a Roman dominating function is the value $$f(V(G))=\sum _{u \in V}f(u)$$. The Roman domination number of a graph G, denoted by $$\gamma _{R}(G)$$, is the minimum weight of a Roman dominating function on G. A connected (respectively, total) Roman dominating function is an RDF f such that the vertices with non-zero labels under f induce a connected graph (respectively, a subgraph with no isolated vertex). The connected (respectively, total) Roman domination number of a graph G, denoted by $$\gamma _{cR}(G)$$ (respectively, $$\gamma _{tR}(G)$$) is the minimum weight of a connected (respectively, total) RDF of G. It this paper we first study the complexity issue of the problems posed in [H. Abdollahzadeh Ahangar, M. A. Henning, V. Samodivkin and I. G. Yero, Total Roman domination in graphs, Appl. Anal. Discret. Math. 10 (2016), 501–517], and show that the problem of deciding whether $$\gamma _{tR}(G)=2\gamma (G)$$, $$\gamma _{tR}(G)=2\gamma _t(G)$$ or $$\gamma _{tR}(G)=3\gamma (G)$$ is NP-hard even when restricted to chordal or bipartite graphs. Then, we give a linear algorithm that decides whether $$\gamma _{tR}(G)=2\gamma (G)$$, $$\gamma _{tR}(G)=2\gamma _t(G)$$ or $$\gamma _{tR}(G)=3\gamma (G)$$, if G is a tree or a unicyclic graph.

Total Roman domination in digraphs

Let D be a finite and simple digraph with vertex set V (D). A Roman dominating function (RDF) on a digraph D is a function f : V (D) → {0, 1, 2} satisfying the condition that every vertex v with f (v) = 0 has an in-neighbor u with f (u) = 2. The weight of an RDF f is the value . An RDF f on D with no isolated vertex is called a total Roman dominating function if the subdigraph of D induced by the set {v ∈ V (D): f (v) ≠ 0} has no isolated vertex. The Roman domination number is the minimum weight of a total Roman dominating function on D. In this paper, we initiate the study of the total Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some sharp bounds for the total Roman domination number and we determine the total Roman domination number of some classes of digraphs.

On the complexity of some hop domination parameters

A hop Roman dominating function (HRDF) on a graph G = ( V , E ) is a function f : V → {0, 1, 2} having the property that for every vertex v ∈ V with f ( v ) = 0 there is a vertex u with f ( u ) = 2 and d ( u , v ) = 2 . The weight of an HRDF f is the sum of its values on V . The minimum weight of an HRDF on G is called the hop Roman domination number of G . An HRDF f is a hop Roman independent dominating function (HRIDF) if for any pair v , w with f ( v ) > 0 and f ( w ) > 0 , d ( v , w ) ≠ 2 . The minimum weight of an HRIDF on G is called the hop Roman independent domination number of G . In this paper, we study the complexity of the hop independent dominating problem, the hop Roman domination function problem and the hop Roman independent domination function problem, and show that the decision problem for each of the above three problems is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.

Roman Free Functions in Graphs

In this paper we introduce a new concepts namely Roman Free Functions, Maximal Roman Free Functions, Roman Free Number and others. Every Maximal Roman Free Function is a Roman Dominating Function. A Minimal Roman Dominating Function need not be a Maximal Roman Free Function. We also give an example of a Maximal Roman Free Function which is not a Minimal Roman Dominating Function. In addition to this we prove that the minimum weight of a Maximal Roman Free Function of a Roman Graph is the Roman Domination Number of the graph.

Some improved inequalities related to Vizing's conjecture

Let γ(G) be the domination number of a simple graph G and G□H be the Cartesian product of two simple graphs G and H. A function f:V(G)→{0,1,2} is a Roman dominating function (RDF) if for each vertex u∈V0,NG(u)∩V2≠∅, where Vi={u∈V(G):f(u)=i}. The Roman domination number γR(G) is the minimum weight f(V(G))=∑u∈V(G)f(u) among all RDFs of G. Vizing conjectured in 1963 that γ(G□H)≥γ(G)γ(H) for any graphs G and H. To this day, this conjecture remains open. In this paper, we show that for each pair of simple graphs G and H, γ(G□H)≥14γR(G)γR(H). This means that Vizing's conjecture holds for any pair of Roman graphs G and H. Moreover, we prove γR(G□H)≥γ(G)γ(H)+12min⁡{γ(G),γ(H)} if G or H is nonempty, which is a slight improvement of γR(G□H)≥γ(G)γ(H) obtained by Wu in 2013 [22].