Roman Domination Number Research Articles

On the perfect differential of a graph

Let G be a graph of order n(G) and vertex set V(G). Given a set S ⊆ V(G), we define the perfect neighbourhood of S as the set Np (S) of all vertices in V(G)\\S having exactly one neighbour in S. The perfect differential of S is defined to be ∂p (S) = |Np (S)| − |S|. In this paper, we introduce the study of the perfect differential of a graph, which we define as ∂p (G) = max{∂p (S) : S ⊆ V(G)}. Among other results, we obtain general bounds on ∂p (G) and we prove a Gallai-type theorem, which states that ∂p (G) + γp R (G) = n(G), where γp R (G) denotes the perfect Roman domination number of G. As a consequence of the study, we show some classes of graphs satisfying a conjecture stated by Bermudo [Discrete Appl. Math. 232 (2017), 64-72].

Algorithmic Aspects of Outer-Independent Total Roman Domination in Graphs

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

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 $, n\not \equiv 0\pmod 5$ .

Roman domination in direct product graphs and rooted product graphs

<abstract><p>Let $ G $ be a graph with vertex set $ V(G) $. A function $ f:V(G)\rightarrow \{0, 1, 2\} $ is a Roman dominating function on $ G $ if every vertex $ v\in V(G) $ for which $ f(v) = 0 $ is adjacent to at least one vertex $ u\in V(G) $ such that $ f(u) = 2 $. The Roman domination number of $ G $ is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all Roman dominating functions $ f $ on $ G $. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination number, and the Roman domination number of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.</p></abstract>

Domination, Edge Domination and Roman Domination in Human Chain Graph

In this paper, we determine domination number, edge domination number and roman domination number of a human chain graph.

A note on the bounds of Roman domination numbers

<abstract> Let $ G $ be a graph and $ f: V(G) \rightarrow \{0, 1, 2\} $ be a mapping. $ f $ is said to be a Roman dominating function of $ G $ if every vertex $ u $ for which $ f(u) = 0 $ is adjacent to at least one vertex $ v $ for which $ f(v) = 2 $. The weight $ w(f) $ of a Roman dominating function $ f $ is the value $ w(f) = \sum_{u \in V(G)}f(u) $, and the minimum weight of a Roman dominating function is the Roman domination number $ \gamma_R(G) $. $ f $ is said to be a Roman $ \{2\} $-dominating function of $ G $ if $ \sum_{v\in N(u)}f(v)\geq 2 $ for every vertex $ u $ with $ f(u) = 0 $, where $ N(u) $ is the set of neighbors of $ u $ in $ G $. The weight of a Roman {2}-dominating function $ f $ is the sum $ \sum_{v\in V}f(v) $ and the minimum weight of a Roman {2}-dominating function is the Roman {2}-domination number $ \gamma_{\{R2\}}(G) $. Chellali et al. (2016) proved that $ \gamma_{R}(G)\geq\frac{\Delta+1}{\Delta}\gamma(G) $ for every nontrivial connected graph $ G $ with maximum degree $ \Delta $. In this paper, we generalize this result on nontrivial connected graph $ G $ with maximum degree $ \Delta $ and minimum degree $ \delta $. We prove that $ \gamma_{R}(G)\geq\frac{\Delta+2\delta}{\Delta+\delta}\gamma(G) $, which also implies that $ \frac{3}{2}\gamma(G)\leq\gamma_{R}(G)\leq 2\gamma(G) $ for any nontrivial regular graph. Moreover, we prove that $ \gamma_{R}(G)\leq2\gamma_{\{R2\}}(G)-1 $ for every graph $ G $ and there exists a graph $ I_k $ such that $ \gamma_{\{R2\}}(I_k) = k $ and $ \gamma_{R}(I_k) = 2k-1 $ for any integer $ k\geq2 $. </abstract>

Algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

Let $$G=(V,E)$$ be a graph. A function $$f : V \rightarrow \{0, 1, 2\}$$ is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex $$v \in V$$ with $$f (v) = 0$$ there is a vertex u adjacent to v with $$f (u) = 2$$ and $$\{x\in V:f(x)=0\}$$ is an independent set. The weight of f is the value $$ f(V)=\sum _{v\in V}f(v)$$ . An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every $$v\in V$$ with $$f(v)>0$$ there is a vertex u adjacent to v with $$f (u)>0$$ . The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by $$\gamma _{oiR}(G)$$ . Similarly, the outer independent total Roman domination number of G is defined, denoted by $$\gamma _{oitR}(G)$$ . In this paper, we first show that computing $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph $$G=(V,E)$$ we propose an algorithm to compute $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) in $${\mathcal {O}}(|V| )$$ time.

Weak Roman domination in rooted product graphs

In this paper, we obtain closed formulae for the weak Roman domination number of rooted product graphs. As a consequence of the study, we show that the use of rooted product graphs is a useful tool to show that the problem of computing the weak Roman domination number of a graph is NP-hard.

Vertex-Edge Roman Domination

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V,E) is a function f : V (G) →{0, 1, 2} such that for each edge e = uv either max{f(u),f(v)}≠0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We first show that determining the number γveR(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then γveR(T) ≥ (n − l − s + 3)∕2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with γveR(T) = 2γ′(T), where γ′(T) is the edge domination number of T.

Bounds on the total double Roman domination number of graphs

Bounds on the total double Roman domination number of graphs

特殊图类的符号罗马控制数

特殊图类的符号罗马控制数

Some progress on the mixed roman domination in graphs

Let G = (V, E) be a simple graph with vertex setxs V and edge set E. A mixed Roman dominating function of G is a function f : V ∪ E → {0, 1, 2} satisfying the condition that every element x ∈ V ∪ E for which f(x) = 0 is adjacent or incident to at least one element y ∈ V ∪ E for which f(y) = 2. The weight of a mixed Roman dominating function f is ω(f) = ∑x∈V∪E f(x). The mixed Roman domination number γR(G) of G is the minimum weight of a mixed Roman dominating function of G. We first show that the problem of computing γR*(G) is NP-complete for bipartite graphs and then we present upper and lower bounds on the mixed Roman domination number, some of them are for the class of trees.

The Roman domination number of some special classes of graphs - convex polytopes

In this paper we study the Roman domination number of some classes of planar graphs - convex polytopes: An, Rn and Tn. We establish the exact values of Roman domination number for: An, R3k, R3k+1, T8k, T8k+2, T8k+3, T8k+5 and T8k+6. For R3k+2, T8k+1, T8k+4 and T8k-1 we propose new upper and lower bounds, proving that the gap between the bounds is 1 for all cases except for the case of T8k+4, where the gap is 2.

Edge lifting and Roman domination in graphs

Let [Formula: see text] be a graph, and let [Formula: see text] be an induced path centered at [Formula: see text]. An edge lift defined on [Formula: see text] is the action of removing edges [Formula: see text] and [Formula: see text] while adding the edge [Formula: see text] to the edge set of [Formula: see text]. In this paper, we initiate the study of the effects of edge lifting on the Roman domination number of a graph, where various properties are established. A characterization of all trees for which every edge lift increases the Roman domination number is provided. Moreover, we characterize the edge lift of a graph decreasing the Roman domination number, and we show that there are no graphs with at most one cycle for which every possible edge lift can have this property.

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 trees with domination number equal to edge-vertex roman domination number

An edge-vertex Roman dominating function (or just ev-RDF) of a graph [Formula: see text] is a function [Formula: see text] such that for each vertex [Formula: see text] either [Formula: see text] where [Formula: see text] is incident with [Formula: see text] or there exists an edge [Formula: see text] adjacent to [Formula: see text] such that [Formula: see text]. The weight of a ev-RDF is the sum of its function values over all edges. The edge-vertex Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of an ev-RDF [Formula: see text]. We provide a characterization of all trees with [Formula: see text], where [Formula: see text] is the domination number of [Formula: see text]

Trees with vertex-edge roman domination number twice the domination number minus one

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V, E) is a function f : V (G) → {0, 1, 2} such that for each edge e = uv either max{f (u), f (v)} ≠ 0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f (w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. We characterize trees with vertexedge roman domination number equal to twice domination number minus one.

Trees with No Locating Roman Domination Critical Vertices

A Roman dominating function (or just RDF) 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 $$f(v) = 2$$ . The weight of an RDF f is the value $$f(V(G))=\sum _{u \in V(G)}f(u)$$ . An RDF f can be represented as $$f=(V_0,V_1,V_2)$$ , where $$V_i=\{v\in V:f(v)=i\}$$ for $$i=0,1,2$$ . An RDF $$f=(V_0,V_1,V_2)$$ is called a locating Roman dominating function (or just LRDF) if $$N(u)\cap V_2\ne N(v)\cap V_2$$ for any pair u, v of distinct vertices of $$V_0$$ . The locating Roman domination number $$\gamma _R^L(G)$$ is the minimum weight of an LRDF of G. A vertex v of a graph G is called a locating Roman domination critical vertex (or just $$\gamma _R^L$$ critical vertex) if $$\gamma _R^L(G-v)<\gamma _R^L(G)$$ . In this paper, we characterize all trees with no $$\gamma _R^L$$ critical vertices.

Properties of double Roman domination on cardinal products of graphs

Double Roman domination is a stronger version of Roman domination that doubles the protection. The areas now have 0 , 1 , 2 or 3 legions. Every attacked area needs 2 legions for its defence, either their own, or borrowed from 1 or 2 neighbouring areas, which still have to keep at least 1 legion to themselves. The minimal number of legions in all areas together is equal to the double Roman domination number. In this paper we determine an upper bound and a lower bound for double Roman domination numbers on cardinal product of any two graphs. Also we determine the exact values of double Roman domination numbers on P 2 × G (for many types of graph G ). Also, the double Roman domination number is found for P 2 × P n , P 3 × P n , P 4 × P n , while upper and lower bounds are given for P 5 × P n and P 6 × P n . Finally, we will give a case study to determine the efficiency of double protection. We will compare double Roman domination versus Roman domination by running a simulation of a battle.

On the Outer-Independent Roman Domination in Graphs

Let G be a graph with no isolated vertex and f:V(G)→{0,1,2} a function. Let Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. The function f is an outer-independent Roman dominating function on G if V0 is an independent set and every vertex in V0 is adjacent to at least one vertex in V2. The minimum weight ω(f)=∑v∈V(G)f(v) among all outer-independent Roman dominating functions f on G is the outer-independent Roman domination number of G. This paper is devoted to the study of the outer-independent Roman domination number of a graph, and it is a contribution to the special issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we obtain new tight bounds for this parameter, and some of them improve some well-known results. We also provide closed formulas for the outer-independent Roman domination number of rooted product graphs.