Note on Strong Roman Domination in Graphs

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 11, HIKARI Ltd, Note on Strong Roman Domination in Graphs Jiaxue Xu and Zhiping Wang Department of Mathematics, Dalian Maritime University Dalian, P. R.China Copyright 2018 Jiaxue Xu and Zhiping Wang. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A strong Roman dominating function (StRDF) for a graph G = (V, E), is a function f : V (G) {0, 1,..., 2 + 1} such that every vertex v B 0 has a neighbor w, and w B 2 and f(w) N(w) B 0, where B j = {v V : f(v) = j} for j = 0, 1 and B 2 = V \(B 0 B1 ) = {v V : f(v) 2}. The strong Roman domination number γ StR (G) of G is the minimum of v V (G) f(v) over such functions. A strong Roman dominating function of G of weight γ StR (G) is called a γ StR (G)-function. M.P.Alvarez et al. [T. Mediavilla- Gradolph, S.M. Sheikholeslami, J.C. Valenzuela-Tripodoro, I.G. Yero, On the strong Roman domination number of graphs, Discrete Applied Mathematics 21(201)44-59] posed the following question: It is true that γ StR (G) 6n for any connected graph G of order n? We solved this part of the problem. In this paper, we prove that strong Roman domination number of some special graphs satisfying γ StR (G) 6n, such as middle graph of fan graph, graphs formed by meridian and latitude lines on a sphere and graph D p,cq, among others. Mathematics Subject Classification: 05C69 Keywords: strong Roman domination number, graph D p,cq, middle graph of fan graph 1 Introduction The definition of a Roman dominating function was motivated by an article in Scientific American by Ian Stewart entitled Defend the Roman Em- Corresponding author

2 56 Jiaxue Xu and Zhiping Wang pire. Cockayne, Dreyer, Hedetnieni, and Hedetniemi [1] defined a Roman dominating function on a graph G = (V, E) to be 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. The weight of an RDF is the value f(v (G)) = u V (G) f(u). The Roman domination number is the minimum weight of an RDF in G. The definition of a Roman dominating function is given implicitly in [2] and []. There are some known results regarding Roman domination number (see Refs. [4-]). Meanwhile, many researches initiate the study of a new parameters related to Roman domination number and begin with the study of several mathematical properties of this variants. For instance, we mention variants like the following ones: strong Roman domination [8], perfect Roman domination [9], Roman 2-domination [10], independent 2-rainbow domination [11], weak Roman domination [], among others. Although the Roman domination strategy can deal with some simple situation, but if several simultaneous attacks to weak places occur, a single stronger place will be not able to defend its neighbors efficiently, thus they fails against a multiple attack situation. With this motivation in mind, the strong Roman domination in graphs was introduced by M.P.Alvarez et al. A graph of order and maximum degree. Let B j = {v V : f(v) = j} for j = 0, 1 and B 2 = V \(B 0 B1 ) = {v V : f(v) 2}. A strong Roman dominating function (StRDF for short) on a graph G = (V, E) is a function f : V (G) {0, 1,..., 2 + 1} satisfying the condition that every vertex v B 0 has a neighbor w, such that w B 2 and f(w) N(w) B 0. Comparing with Roman dominating, strong Roman domination defensive strategy needs more legions than it, so the advantage is not to save resources but to design a stronger empire against external attacks. Under the strong Roman dominating strategy, any strong vertex must be able to defend itself and at least one half of its weak neighbors. At the same time, M.P.Alvarez et al show that the decision problem regarding the computation of the strong Roman domination number is NP-complete, even when restricted to bipartite graphs. Moreover, they obtain several bounds on such a parameter and give some realizability results for it. For instances, they prove that for any tree T of order n. The present paper is organized as follows. In Section 2, we list some necessary notation we shall use throughout this paper. In Section, we prove that strong Roman domination number of some special graphs satisfying γ StR (G) 6n, such as middle graph of fan graph, graphs formed by meridian and latitude lines on a sphered and graph D p,cq, among others.

3 Note on strong Roman domination in graphs 5 2 Notation For notation and graph theory terminology in general we follow [1]. To begin with our work, we first introduce the terminology and notation we shall use throughout this paper. All graphs considered in this paper are undirected, finite and simple. Connectivity is perhaps the most fundamental graph-theoretic property. G is a simple graph with vertex set V (G) and edge set E(G) (briefly V and E). The order V of G is denoted by n and the order E of G is denoted by m. For every vertex v V, the open neighborhood N(v) is {u V (G) uv E(G)} and the closed neighborhood of V is the set N[v] = N(v) {v}. If S is a subset of V (G) then N(S) = v S N(v), N[S] = N(S) S and the subgraph induced by S in G is denoted G[S]. Respectively, the degree of v is d(v) = N(v). By δ = δ(g), we denote the maximum degree of the graph G. Main results The graph D p,cq [10] consists of p cycles with one common vertex, which denoted by v 1. And each cycle has q vertices besides the center point v 1. The principle of labelling vertices of the D p,cq graph is the first of the common vertex is labeled, the following each cycle is labelled in proper sequence. See Figure 1. Figure 1: The graph D p,c6 Theorem.1. Let D p,cq graph with order n. Then γ StR (D p,cq ) 6n. Proof. It is straightforward to observe that the number of vertices are n = (q 1)p+1 and m = pq edges. Then the average of degree is d = 2pq. The (q 1)p+1 graph D p,cq N[v 1 ] consists of p paths P q, and we let g(x) is γ StR (P q ) function, such that x V (P q ). So we can draw the strong Roman domi-

4 58 Jiaxue Xu and Zhiping Wang nation number of each paths P q is γ StR (P q ) = w(g(x)) = 2(q ). Now define a function f : V (G) {0, 1,..., (G) + 1} on G as follows: f(v 2 1 ) = 1+ 2p = 1+p, f(u) = 0 for every vertex u N(v 2 1), f(x) = g(x) to remaining vertexes. Clearly f is a strong Roman dominating function of graph D p,cq and implies thst 2(q ) γ StR (D p,cq ) w(f) = p p 2(q ) p + p = 1 + 2pq 2p < 6n. Therefore, the proof is complete. The principle of labelling vertices of the spider web graph W (p, q) is the first of the common vertex is labeled, the following each cycle is labelled in proper sequence. See Figure 2. Figure 2: The spider web graph W(2,4) Theorem.2. Let W (p, q) a spider web graph with order n, Then γ StR (W (p, q)) 6n. Proof. Observe that V (W (p, q)) = q(p + 1) + 1. Consider the following five subcases. Case 1: Let p 1, q = 1. It is obvious that graph W (p, q) is a path. According to theorem previously, we can obtain that γ StR (W (p, q) < 6n. Case 2 :Let p = 1, q = 2. There are n = 5. Now we define f : V (G) {0, 1,..., + 1} on V (W (p, q)) as follows: f(x 2 0) = 2, f(u) = 0 for each vertexes u N(x 0 ), and f(v) = 1 to remaining vertexes. Clearly f is a StRDF of W (p, q) of weight w(f) = = 4. Using the fact that γ StR (W (p, q)) w(f) = 4 < 0.

5 Note on strong Roman domination in graphs 59 Case : Let p = 1, q. There are n = 2q + 1. Now we define f : V (G) {0, 1,..., + 1} on V (W (p, q)) as follows: f(x 2 0) = 1 + q, f(u) = 0 for 2 each vertexes u N(x 0 ), and f(v) = 1 to remaining vertexes. Therefore f is a StRDF of W (p, q) of weight w(f) = 1 + q + q, and we deduce that 2 γ StR (W (p, q)) w(f) = 1 + q 2 + q 1 + q q 2 = q < 6n. Case 4 : Let p = 2, q 2. There are n = q + 1. Now we define f : V (G) {0, 1,..., +1} on V (W (p, q)) as follows: f(v) = 2 for every vertexes v C 2 q 2, f(u) = 0 for each vertexes u N(v), and f(x 0 ) = 1. Obviously f is a StRDF of W (p, q) of weight w(f) = 1 + 2q and γ StR (W (p, q)) w(f) = 1 + 2q < 6n. Case 5 : Let p, q 2. According to the definition of spider web graph, there are p circle C q. We name these circles. From inside to outside, the circles were named as C q1, C q2,..., C qp. Now we define f : V (G) {0, 1,..., + 1} 2 on V (W (p, q)) as follows: f(v) = 2 for each vertexes v C qp, f(x) = 0 to vertices x N(v), f(u) = 1 + q to x 2 0, and f(x) = 0 to every vertices x q 1. We can obtain that γ StR (C q ) = 2q, so we have C q 2 =... = C qp 2 = 2q. Clearly f is a StRDF of W (p, q) of weight w(f) = 2q + (p ) 2q q 2 and we deduce that Because γ StR (W (p, q)) w(f) = 2q + (p ) 2q q 2 2q + (p )( 2q + 1 ) + n + 2 = 2pq + p + q < 6n. 6n 2pq + p + q = 4pq q 14 p 9 14 = ( 4n )m + 5n > 0. Therefore, the proof is complete. The principle of labelling vertices of the middle graph of fan graph M(F p ) is the first of the common vertex is labeled, the following each vertexes is labelled in proper sequence. See Figure. Theorem.. Let M(F p ) a middle graph of fan graph with order n, Then γ StR (M(F p )) 6n. Proof. Observe that V (M(F p )) = p. Let f(v 0 ) = 1 + p 1 2 and f(v 01) =

6 540 Jiaxue Xu and Zhiping Wang Figure : The middle graph of fan graph 1+ p 2, 0 to N(v 2 01) N(v 0 ). We can see the remaining vertexes as a P (p 2) path. Assuming g(x) is the γ StR (P (p 2)) function, and we let f(x) = g(x) to remaining vertexes. Clearly f is a StRDF of W (p, q) of weight w(f) = 1 + p p 2 + 2p 4. And we have 2 2 γ StR (M(F p )) w(f) = 1 + p p p p p 1 = 5p < 6n. It follows that 18p 18 5p 1 = 19p 2 Consequently, the proof is complete. 4 Conclusion M. P. Alvarez et al. posed the following question: It is true that γ StR (G) 6n for any connected graph G of order n? We should find more special graphs to prove it or search some graphs overturn the conclusion. References [1] Ernie J. Cockayne, Paul A. Dreyer, Sandra M. Hedetniemi, Stephen T. Hedetniemi, Roman domination in graphs, Discrete Mathematics, 28 (2000), no. 1, [2] C. S. Revelle, K. E. Rosing, Defendens Imperium Romanum: A Classical Problem in Military Strategy, American Mathematical Monthly, 10 (2000),

7 Note on strong Roman domination in graphs 541 no., [] I. Stewart, Defend the Roman Empire!, Scientific American, 281 (1999), no. 6, [4] M. A. Henning, S. T. Hedetniemi, Defending the Roman Empire-A new strategy, Discrete Mathematics, 266 (200), no. 1, [5] M. A. Henning, Defending the Roman Empire from Multiple Attacks, Elsevier Science Publishers B. V [6] O. Favaron, H. Karami, R. Khoeilar, S.M. Sheikholeslami, On the Roman domination number of a graph, Discrete Mathematics, 09 (2009), no. 10, [] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, [8] M. P. lvarez-ruiz, T. Mediavilla-Gradolph, S. M. Sheikholeslami, J.C. Valenzuela-Tripodoro, I.G. Yero, On the strong Roman domination number of graphs, Discrete Applied Mathematics, 21 (201), [9] M. A. Henning, W. F. Klostermeyer, G. Macgillivray, Perfect Roman domination in trees, Discrete Applied Mathematics, 26 (2018), [10] M. Chellali, T. W. Haynes, S. T. Hedetniemi, Alice A. McRae, Roman 2-domination, Discrete Applied Mathematics, 2014 (2016), [11] A. Rahmouni, M. Chellali, Independent Roman 2 -domination in graphs, Discrete Applied Mathematics, 26 (2018), Received: April 2, 2018; Published: May 15, 2018