Roman Domination Number Of Graphs Research Articles

On maximal Roman domination in graphs: complexity and algorithms

For a simple undirected connected graph G = (V,E), a maximal Roman dominating function (MRDF) of G is a function f : V (G) → {0,1,2} with the following properties: (i) For every vertex v ∈ {v ∈ V |f(v) = 0}, there exists a vertex u ∈ N(v) such that f(u) = 2. (ii) The set {v ∈ V |f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v ∈ {v ∈ V |f(v) = 0} such that N(v) ∩ {u ∈ V |f(u) = 0} = ∅. The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f(G) =Σv∈V (G)f(v), and the maximal Roman domination number of G, denoted by γmR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.

On the signed strong total Roman domination number of graphs

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed strong total Roman dominating function ona graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ forwhich $f(v)=-1$ is adjacent to at least one vertex$w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where$V_{-1}=\{v\in V: f(v)=-1\}$.The minimum of thevalues $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strongtotal Roman dominating functions $f$ of $G$, is called the signed strong totalRoman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$.In this paper, we initiate signed strong total Roman domination number of a graph and giveseveral bounds for this parameter. Then, among other results, we determine the signed strong total Roman dominationnumber of special classes of graphs.

New results on quadruple Roman domination in graphs

Let [Formula: see text] be an integer and [Formula: see text] be a simple graph with vertex set [Formula: see text]. Let [Formula: see text] be a function that assigns label from the set [Formula: see text] to the vertices of a graph [Formula: see text]. For a vertex [Formula: see text], the active neighborhood of [Formula: see text], denoted by [Formula: see text], is the set of vertices [Formula: see text] such that [Formula: see text]. A [Formula: see text]-RDF is a function [Formula: see text] satisfying the condition that for any vertex [Formula: see text] with [Formula: see text], [Formula: see text]. The weight of a [Formula: see text]-RDF is [Formula: see text]. The [Formula: see text]-Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an [Formula: see text]-RDF on [Formula: see text]. The case [Formula: see text] is called quadruple Roman domination number. In this paper, we first establish an upper bound for quadruple Roman domination number of graphs with minimum degree two, and then we derive a Nordhaus–Gaddum bound on the quadruple Roman domination number of graphs.

Further results on the independent Roman domination number of graphs

Let f : V (G) → {0, 1, 2} be a function on a graph G with vertex set V (G). Let Vi = {v ∈ V (G) : f (v) = i} for every i ∈ {0, 1, 2}. The function f is said to be an independent Roman dominating function on G if V 1 ∪ V 2 is an independent set and for every υ 2 V 0. The minimum weight among all independent Roman dominating functions f on G is the independent Roman domination number of G, and is denoted by iR (G). In this paper we continue with the study of this parameter. In particular, we provide new bounds on iR (G) in terms of other domination invariants. Some of our results are tight bounds that improve some well-known results. Finally, we compute the independent Roman domination number of some product graphs, and we provide an alternative proof to show that the problem of computing iR (G) is NP-hard.

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 .

Maximal first Zagreb index of trees with given Roman domination number

<abstract><p>The first Zagreb index of graphs is defined to be the sum of squares of degrees of all the vertices of graphs. It drew a great deal of attention in the past half-century. In this paper, we study the relationship between the first Zagreb index and Roman domination number of graphs. More precisely, we characterize the trees with the maximum first Zagreb index among trees with given Roman domination number.</p></abstract>

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.

Quadruple Roman domination in graphs

Let [Formula: see text] be a finite and simple graph with vertex set [Formula: see text]. Let [Formula: see text] be a function that assigns label from the set [Formula: see text] to the vertices of a graph [Formula: see text]. For a vertex [Formula: see text], the active neighborhood of [Formula: see text], denoted by [Formula: see text], is the set of vertices [Formula: see text] such that [Formula: see text]. A quadruple Roman dominating function (QRDF) is a function [Formula: see text] satisfying the condition that for any vertex [Formula: see text] with [Formula: see text]. The weight of a QRDF is [Formula: see text]. The quadruple Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a QRDF on [Formula: see text]. In this paper, we investigate the properties of the quadruple Roman domination number of graphs, present bounds on [Formula: see text] and give exact values for some graph families. In addition, complexity results are also obtained.

Bounds on the total double Roman domination number of graphs

Bounds on the total double Roman domination number of graphs

On the signed strong Roman domination number of graphs

Let [Formula: see text] be a finite and simple graph of order [Formula: see text] and maximum degree [Formula: see text]. A signed strong Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) for every vertex [Formula: see text] of [Formula: see text], [Formula: see text], where [Formula: see text] is the closed neighborhood of [Formula: see text] 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], where [Formula: see text]. The minimum of the values [Formula: see text], taken over all signed strong Roman dominating functions [Formula: see text] of [Formula: see text], is called the signed strong Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we continue the study signed strong Roman domination number of a graph and give several bounds for this parameter. Then, among other results, we determine the signed strong Roman domination number of special classes of graphs.

English

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 the 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 the vertex v must have at least one neighbor with f (w) ⩾ 2. The weight of a double Roman dominating function f is the sum $$f(V) = \sum\nolimits_{v \in V} {f(v)} $$. The minimum weight of a double Roman dominating function on G is called the double Roman domination number of G and is denoted by γdR(G). In this paper, we establish a new upper bound on the double Roman domination number of graphs. We prove that every connected graph G with minimum degree at least two and G ≠ C5 satisfies the inequality $${\gamma _{{\rm{dR}}}}(G)\leqslant \left\lfloor {{\textstyle{{13} \over {11}}}n} \right\rfloor $$. One open question posed by R. A. Beeler et al. has been settled.

Quasi-total Roman Domination in Graphs

A quasi-total Roman dominating function on a graph G= (V, E) is a function f: V→ { 0 , 1 , 2 } satisfying the following:Every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 , andIf x is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then f(x) = 1. The weight of a quasi-total Roman dominating function is the value ω(f) = f(V) = ∑ u ∈ Vf(u). The minimum weight of a quasi-total Roman dominating function on a graph G is called the quasi-total Roman domination number of G. We introduce the quasi-total Roman domination number of graphs in this article, and begin the study of its combinatorial and computational properties.

Note on the perfect Roman domination number of graphs

In this note, we list some propositions of the perfect Roman domination number of graphs and give the characterization of graphs G with special value of the perfect Roman domination. Furthermore, γRp(F)+γRp(F¯) is given. At the last, a linear time algorithm is shown to compute the perfect Roman domination number of a cograph.

An improved upper bound on the double Roman domination number of graphs with minimum degree at least two

A double Roman dominating function (DRDF) on a graph G=(V,E) 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 w with f(w)=3, and if f(v)=1, then vertex v must have at least one neighbor w with f(w)≥2. The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number γdR(G) is the minimum weight of a DRDF on G. Let G be a connected graph G of order n and minimum degree at least two. With the exception of seven graphs of order at most seven, Beeler et al. (2016) observed that γdR(G)≤6n5 and posed the question whether this bound can be improved. Amjadi et al. (2018) settled this question by proving that γdR(G)≤8n7. In this paper, we improve this bound to γdR(G)≤11n10. Moreover, we provide an infinite family of graphs attaining this bound.

Independent Roman Domination Number of Graphs

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On the weak Roman domination number of lexicographic product graphs

A vertex v of a graph G=(V,E) is said to be undefended with respect to a function f:V⟶{0,1,2} if f(v)=0 and f(u)=0 for every vertex u adjacent to v. We call the function f a weak Roman dominating function if for every v such that f(v)=0 there exists a vertex u adjacent to v such that f(u)∈{1,2} and the function f′:V⟶{0,1,2} defined by f′(v)=1, f′(u)=f(u)−1 and f′(z)=f(z) for every z∈V∖{u,v}, has no undefended vertices. The weight of f is w(f)=∑v∈V(G)f(v). The weak Roman domination number of a graph G, denoted by γr(G), is the minimum weight among all weak Roman dominating functions on G. Henning and Hedetniemi (2003) showed that the problem of computing γr(G) is NP-Hard, even when restricted to bipartite or chordal graphs. This suggests finding γr(G) for special classes of graphs or obtaining good bounds on this invariant. In this article, we obtain closed formulae and tight bounds for the weak Roman domination number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product.

On the strong Roman domination number of graphs

Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a “stronger” neighbor place (having two legions), a graph theoretical model called Roman domination in graphs was described. A Roman dominating function for a graph G=(V,E), is a function f:V→{0,1,2} such that every vertex v with f(v)=0 has at least a neighbor w in G for which f(w)=2. The Roman domination number of a graph is the minimum weight, ∑v∈Vf(v), of a Roman dominating function. In this paper we initiate the study of a new parameter related to Roman domination, which we call strong Roman domination number and denote it by γStR(G). We approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. In particular, we first show that the decision problem regarding the computation of the strong Roman domination number is NP-complete, even when restricted to bipartite graphs. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, we prove that for any tree T of order n≥3, γStR(T)≤6n/7 and characterize all extremal trees.

Signed total Roman domination in graphs

Let $$G$$G be a finite and simple graph with vertex set $$V(G)$$V(G). A signed total Roman dominating function (STRDF) on a graph $$G$$G is a function $$f:V(G)\rightarrow \{-1,1,2\}$$f:V(G)ź{-1,1,2} satisfying the conditions that (i) $$\sum _{x\in N(v)}f(x)\ge 1$$źxźN(v)f(x)ź1 for each vertex $$v\in V(G)$$vźV(G), where $$N(v)$$N(v) is the neighborhood of $$v$$v, and (ii) every vertex $$u$$u for which $$f(u)=-1$$f(u)=-1 is adjacent to at least one vertex $$v$$v for which $$f(v)=2$$f(v)=2. The weight of an SRTDF $$f$$f is $$\sum _{v\in V(G)}f(v)$$źvźV(G)f(v). The signed total Roman domination number $$\gamma _{stR}(G)$$źstR(G) of $$G$$G is the minimum weight of an STRDF on $$G$$G. In this paper we initiate the study of the signed total Roman domination number of graphs, and we present different bounds on $$\gamma _{stR}(G)$$źstR(G). In addition, we determine the signed total Roman domination number of some classes of graphs.

Unique response Roman domination in graphs

A function f : V ( G ) → { 0 , 1 , 2 } is a Roman dominating function if every vertex u for which f ( u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2 . A function f : V ( G ) → { 0 , 1 , 2 } with the ordered partition ( V 0 , V 1 , V 2 ) of V ( G ) , where V i = { v ∈ V ( G ) ∣ f ( v ) = i } for i = 0 , 1 , 2 , is a unique response Roman function if x ∈ V 0 implies | N ( x ) ∩ V 2 | ≤ 1 and x ∈ V 1 ∪ V 2 implies that | N ( x ) ∩ V 2 | = 0 . A function f : V ( G ) → { 0 , 1 , 2 } is a unique response Roman dominating function if it is a unique response Roman function and a Roman dominating function. The unique response Roman domination number of G , denoted by u R ( G ) , is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination number of graphs and present bounds for this parameter.