Roman Domination Number Research Articles

On the Unique Response Roman Domination Numbers of Graphs

Let G be a graph with vertex set V(G). A function f: V(G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V(G), where Vi = {V∈V(G) | f(V) = i} for i = 0, 1, 2, is a Roman dominating function if x ∈ V0 implies |N(x)∩V2|≥ 1. It is a unique response Roman function if x ∈ V0 implies |N(x) ≥ V2|≤ 1 and x ∈ V1 ∪ V2 implies that |N(x) ∩ V2| = 0. A function f: V(G) → {0, 1, 2} is a unique response Roman dominating function if it is both a unique response Roman function and a Roman dominating function. The unique response Roman domination number, denoted by uR(G), of G is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination of graphs, and provide some graphs whose unique response Roman domination number equals to the independent Roman domination number.

Twin signed Roman domination numbers in directed graphs

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed Roman dominating function (TSRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. The weight of an TSRDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The twin signed Roman domination number $\gamma_{sR}^*(D)$ of $D$ is the minimum weight of an TSRDF on $D$. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on $\gamma_{sR}^*(D)$. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

Roman domination in graphs: The class ℛUV R

For a graph [Formula: see text], a Roman dominating function (RDF) [Formula: see text] has the property that every vertex [Formula: see text] with [Formula: see text] has a neighbor [Formula: see text] with [Formula: see text]. The weight of a RDF [Formula: see text] is the sum [Formula: see text], and the minimum weight of a RDF on [Formula: see text] is the Roman domination number [Formula: see text] of [Formula: see text]. The Roman bondage number [Formula: see text] of [Formula: see text] is the minimum cardinality of all sets [Formula: see text] for which [Formula: see text]. A graph [Formula: see text] is in the class [Formula: see text] if the Roman domination number remains unchanged when a vertex is deleted. In this paper, we obtain tight upper bounds for [Formula: see text] and [Formula: see text] provided a graph [Formula: see text] is in [Formula: see text]. We present necessary and sufficient conditions for a tree to be in the class [Formula: see text]. We give a constructive characterization of [Formula: see text]-trees using labelings.

On the signed total Roman domination and domatic numbers of graphs

A signed total Roman dominating function on a graph G is a function f:V(G)⟶{−1,1,2} such that ∑u∈N(v)f(u)≥1 for every vertex v∈V(G), where N(v) is the neighborhood of v, and every vertex u∈V(G) for which f(u)=−1 is adjacent to at least one vertex w for which f(w)=2. The signed total Roman domination number of a graph G, denoted by γstR(G), equals the minimum weight of a signed total Roman dominating function. A set {f1,f2,…,fd} of distinct signed total Roman dominating functions on G with the property that ∑i=1dfi(v)≤1 for each v∈V(G), is called a signed total Roman dominating family (of functions) on G. The maximum number of functions in a signed total Roman dominating family on G is the signed total Roman domatic number of G, denoted by dstR(G). In this paper we continue the investigation of the signed total Roman domination number, and we initiate the study of the signed total Roman domatic number in graphs. We present sharp bounds for γstR(G) and dstR(G). In addition, we determine the total signed Roman domatic number of some graphs.

On the Global Roman Domination Number in Graphs

A Roman dominating function \( f \) on a graph \( G \) is a global Roman dominating function on \( G \), if \( f \) is also a Roman dominating function on \( \bar{G} \). The weight of a global Roman dominating function \( f \) is the value \( w(f) = \sum\nolimits_{x \in V(G)} {f(x)} \). The minimum weight of a global Roman dominating function on a graph \( G \) is called the global Roman domination number \( \gamma_{gR} (G) \) of \( G \). In this paper, we present upper bounds for \( \gamma_{gR} (G) \) in terms of order, diameter, and girth. We give necessary and sufficient conditions for a graph \( G \) with property \( \gamma_{gR} (G) = \gamma_{g} (G) + i \) for all \( i = 0,1, 2,3 \), where \( \gamma_{g} (G) \) is the global domination number of \( G \). We also describe all connected unicyclic graphs \( G \) for which \( \gamma_{gR} \left( G \right) - \gamma_{R} (G) \) is maximum.

Double Roman domination

For a graph G=(V,E), a double Roman dominating function is a function f:V→{0,1,2,3} having the property that if f(v)=0, then vertex v must have at least two neighbors assigned 2 under f or one neighbor with f(w)=3, and if f(v)=1, then vertex v must have at least one neighbor with f(w)≥2. The weight of a double Roman dominating function f is the sum f(V)=∑v∈Vf(v), and the minimum weight of a double Roman dominating function on G is the double Roman domination number of G. We initiate the study of double Roman domination and show its relationship to both domination and Roman domination. Finally, we present an upper bound on the double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound.

Strong equality of Roman and weak Roman domination in trees

We provide a constructive characterization of the trees for which the Roman domination number strongly equals the weak Roman domination number, that is, for which every weak Roman dominating function of minimum weight is a Roman dominating function. Our characterization is based on five simple extension operations, and reveals several structural properties of these trees.

Edge Roman Domination on Graphs

An edge Roman dominating function of a graph G is a function $$f:E(G) \rightarrow \{0,1,2\}$$ satisfying the condition that every edge e with $$f(e)=0$$ is adjacent to some edge $$e'$$ with $$f(e')=2$$ . The edge Roman domination number of G, denoted by $$\gamma '_R(G)$$ , is the minimum weight $$w(f) = \sum _{e\in E(G)} f(e)$$ of an edge Roman dominating function f of G. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if G is a graph of maximum degree $$\Delta $$ on n vertices, then $$\gamma _R'(G) \le \lceil \frac{\Delta }{\Delta +1} n \rceil $$ . While the counterexamples having the edge Roman domination numbers $$\frac{2\Delta -2}{2\Delta -1} n$$ , we prove that $$\frac{2\Delta -2}{2\Delta -1} n + \frac{2}{2\Delta -1}$$ is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of k-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on n vertices is at most $$\frac{6}{7}n$$ , which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain $$K_{2,3}$$ as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.

Averaging 2-rainbow domination and Roman domination

For a graph G, let γr2(G) and γR(G) denote the 2-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (2013) proved γr2(G)+γR(G)≤64n(G) for a connected graph G of order n(G) at least 3. Furthermore, they conjectured γr2(G)+γR(G)≤43n(G) for a connected graph G of minimum degree at least 2 that is distinct from C5. We characterize all extremal graphs for their inequality and prove their conjecture.

Roman game domination number of a graph

The Roman game domination number of an undirected graph G is defined by the following game. Players $$\mathcal {A}$$A and $$\mathcal {D}$$D orient the edges of the graph G alternately, with $$\mathcal {D}$$D playing first, until all edges are oriented. Player $$\mathcal {D}$$D (frequently called Dominator) tries to minimize the Roman domination number of the resulting digraph, while player $$\mathcal {A}$$A (Avoider) tries to maximize it. This game gives a unique number depending only on G, if we suppose that both $$\mathcal {A}$$A and $$\mathcal {D}$$D play according to their optimal strategies. This number is called the Roman game domination number of G and is denoted by $$\gamma _{Rg}(G)$$źRg(G). In this paper we initiate the study of the Roman game domination number of a graph and we establish some bounds on $$\gamma _{Rg}(G)$$źRg(G). We also determine the Roman game domination number for some classes of graphs.

Total Roman domination in graphs

A Roman dominating function on a graph G is a function f:V(G) → {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. The weight of a Roman dominating function f is the sum, ΣuV(G) f(u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.

Lower bounds on the Roman and independent Roman domination numbers

A Roman dominating function (RDF) on a graph G is a function f : V (G) ? {0,1,2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the sum f(V) = ?v?V f(v), and the minimum weight of a Roman dominating function f is the Roman domination number ?R(G). An RDF f is called an independent Roman dominating function (IRDF) if the set of vertices assigned positive values under f is independent. The independent Roman domination number iR(G) is the minimum weight of an IRDF on G. We show that for every nontrivial connected graph G with maximum degree ?, ?R(G)? ?+1/??(G) and iR(G) ? i(G) + ?(G)/?, where ?(G) and i(G) are, respectively, the domination and independent domination numbers of G. Moreover, we characterize the connected graphs attaining each lower bound. We give an additional lower bound for ?R(G) and compare our two new bounds on ?R(G) with some known lower bounds.

Twin Roman domination number of a digraph

Twin Roman domination number of a digraph

Roman and Total Domination

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑u ∈ V (G) f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.

Roman domination in subgraphs of grids

In a Roman domination of a graph, vertices are assigned a value from {0,1,2} in such a way that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. The Roman domination number is the minimum possible sum of all values in such an assignment. In this paper, we prove that the Roman domination number is NP-hard even for subgraphs of grids and it is APX-hard even for bipartite graphs with maximum degree 4. We also prove that the Roman domination number is fixed parameter tractable for graphs with bounded local treewidth, as graphs with bounded maximum degree or bounded genus (like planar graphs or toroidal graphs). We also obtain complexity results regarding the number of Roman dominating sets.

Trees with independent Roman domination number twice the independent domination number

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that 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 RDF [Formula: see text] is the value [Formula: see text]. The Roman domination number, [Formula: see text], of [Formula: see text] is the minimum weight of a RDF on [Formula: see text]. An RDF [Formula: see text] is called an independent Roman dominating function (IRDF) if the set [Formula: see text] is an independent set. The independent Roman domination number, [Formula: see text], is the minimum weight of an IRDF on [Formula: see text]. In this paper, we study trees with independent Roman domination number twice their independent domination number, answering an open question.

Global Roman domination in graphs

A Roman dominating function (RDF) on a graph G=(V,E) is defined to be a function f:V→{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. A set S⊆V is a global dominating set if S dominates both G and its complement G¯. The global domination number γg(G) of a graph G is the minimum cardinality of S. We define a global Roman dominating function on a graph G=(V,E) to be a function f:V→{0,1,2} such that f is an RDF for both G and its complement G¯. The weight of a global Roman dominating function is the value f(V)=∑u∈Vf(u). The minimum weight of a global Roman dominating function on a graph G is called the global Roman domination number of G and denoted by γgR(G). In this paper, we initiate a study of this parameter.

The [formula omitted]-valued Roman Domination and Its Boundaries

The Roman dominating function on a graph G=(V,E) is a labeling f:V→{0,1,2} satisfying that any vertex v with f(v)=0 is adjacent to a vertex u with f(u)=2. In this paper, we generalize the notion of independence and dominance between two vertices. This gives a new generalization of Roman domination, called Nk-valued Roman domination, where the codomain of the Roman dominating function is extended to Nk={0,1,2,…,k}. Two lower bounds of this Nk-valued Roman domination number in terms of the diameter and radius of G respectively are established.

On maximal Roman domination in graphs

A Roman dominating function (RDF) for a graph is a function satisfying the condition that every vertex u of G for which is adjacent to at least one vertex v of G for which . The weight of a RDF f is the sum , and the minimum weight of a RDF for G is the Roman domination number, of G. A maximal RDF for a graph G is a RDF f such that is not a dominating set of G. The maximal Roman domination number, of a graph G equals the minimum weight of a maximal RDF for G. We first show that determining the number for an arbitrary graph G is NP-complete even when restricted to bipartite or planar graphs. Then, we characterize connected graphs G such that . We also provide a characterization of all trees T of order n such that.

Mixed Roman Domination in Graphs

Let $$G = (V, E)$$ be a simple graph with vertex set V and edge set E. A mixed Roman dominating function (MRDF) of G is a function $$f: V\cup E\rightarrow \{0,1,2\}$$ satisfying the condition every element $$x\in V\cup E$$ for which $$f(x)= 0$$ is adjacent or incident to at least one element $$y\in V\cup E$$ for which $$f(y) = 2$$ . The weight of a MRDF f is $$\omega (f)=\sum _{x\in V\cup E}f(x)$$ . The mixed Roman domination number of G is the minimum weight of a mixed Roman dominating function of G. In this paper, we initiate the study of the mixed Roman domination number and we present bounds for this parameter. We characterize the graphs attaining an upper bound and the graphs having small mixed Roman domination numbers.