The Split Domination and Irredundant Number of a Graph

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1 The Split Domination and Irredundant Number of a Graph S. Delbin Prema 1, C. Jayaekaran 2 1 Department of Mathematic, RVS Technical Campu-Coimbatore, Coimbatore , Tamil Nadu, India 2 Department of Mathematic, Pioneer Kumarawamy College, Nagercoil , Tamil Nadu, India ABSTRACT For a imple connected graph G and D V = V (G). Here we have continue ome relation between plit domination number and irredundant number. We give a characterization of graph G for which ( G) ir( G) for every graph G. Moreover ome pecial graph, ufficient condition of the domination number ( G) 2. Further, we have developed a method of contruction of a graph with a given number a the plit domination number. Key Word: Irredundant et, Irredundant number, P-N et, Split Domination et, Split Domination number. I. INTRODUCTION Graph theory i one of the mot important branche of modern mathematic and computer application. Here we dicu the relation of plit domination and irredundant number of a graph. For a graph G = (V, E) and a vertex v V. The open neighbourhood of v i the et N(v) = {u V uv E} and the cloed neighbourhood of v i N[v] = N(v) {v}. A dominating et i a et w V for which N[w] = V, or equivalently, for every vertex u V - D, we have N (u) D. A et D of vertice in a graph G i a dominating et of G if every vertex in V - D i adjacent to ome vertex in D. A vertex v in a graph G i aid to dominate itelf and each of it neighbour. We ay with other word, v dominate the vertice of it cloed neighbourhood N[v]. The domination number of G equal the minimum cardinality over all dominating et in G and it i denoted by (G). A dominating et of cardinality (G) i referred to a a minimum dominating et. A dominating et of cardinality (G) i called a - et. A minimal dominating et in a graph G i a dominating et that contain no dominating et a a proper ubet. A minimal dominating et of minimum cardinality i a minimum dominating et and conit of (G) vertice. It wa in [1]. A et D i irredundant if for every vertex u D, we have N [u] N [D {u}]. A ubet D V i irredundant if for each x D, D {x} doe not dominate N [x]. Equivalently, D i an irredundant et of vertice if N [D {v}] N [D] for every 745 P a g e

2 vertex v D. The irredundant number of G, denoted by ir(g), i the minimum cardinality taken over all maximal irredundant et of vertice of G. An irredundant et of cardinality ir(g) i called an ir-et. It wa pecified in [2, 3]. A et D V i a dominating et of G if N [D] = v, while D V i an irredundant et of G if PN (x, D) = N [x] N [D {x}] for every x D. D i a minimal dominating if and only if it i dominating and irredundant. An irredundant et D i maximal irredundant if no proper uperet of D i irredundant. Which are indicated in [4]. Every vertex v with the property N [D {v}] N [D] i an irredundant vertex. Conequently, every vertex in an irredundant et i an irredundant vertex. we refer to [5]. For a imple graph G and we oberve that D i a P-N et if and only if the et of D - perfect vertice i a dominating et of G. It wa pecified in [6]. Theorem 1.1 [1] If G i a graph and D V (G) i a dominating et then D i an irredundant et. The above obervation motivate u to tudy of relation between plit domination and irredundant number. The plit domination and irredundant number of certain tandard clae of graph are determined. Variou characterization reult are proved. II. BASIC DEFINITIONS Definition 2.1 (Kulli and Janakiram[7]) A dominating et D V (G) i a plit dominating et if the induced ubgraph < V \ D > i either diconnected or a K 1. Definition 2.2 [7] A plit domination et D i a minimal plit dominating et if i) every vertex v D ha a private neighbour with repect to D or ii) for every vertex v D, the induced graph < (V \ D) {v} > i connected. Definition 2.3 [7] Let G = (V,E) be a graph. Then (G) = min{ D : D i a plit domination et} i the plit domination number and (G) = max{ D : D i a minimal plit dominating et} i the upper plit domination number. Definition 2.4 [8] A plit irredundant et i a et D uch that i) for u D, u ha a private neighbour with repect to V (S) and ii) the induced graph < V \ D > i either diconnected or K 1. Definition 2.5 [8] A maximal plit irredundant et D i a plit irredundant et uch that for every v V \ Done of the following hold true : i) v doe not have a private neighbour with repect to V (D {v}) or ii) the induced graph < V \ {D v} > i connected Definition 2.6 [8] Let G = (V, E) be a graph. Then ir (G) = min{ D : D i a maximal plit irredundant et} i the lower plit irredundant number and IR(G) = max{ D : D i a maximal plit irredundant et} i the upper plit irredundant number. Definition 2.7 [6] A et D V = V (G), vertex u of G i known a D - perfect if N[u] D = P a g e

3 Definition 2.8 [6] The et D i called a perfect neighbourhood et if for all v V, v or ome neighbour of v i D - perfect. III. IMPORTANT RESULTS Some known inequalitie and bound that are of interet are the following: [7] ( G) ( G) [7] ( G) n. (G) / ( (G) + 1). [9] ( G) ( G) n. [8] For the bipartite graph K m, n ir (G) = (G) = min {m, n}. [8] For a path P n, (G) = IR(G) = max{m, n}. ir (G) = (G) = n /3. (G) = IR(G) = [8] For a cycle C n, n /2. ir (G) = (G) = n /3. (G) = IR(G) = n /2. [8] (C p ) = [p / 3], where [x] i the leat + ve integer not le than x and C p i a cycle with p 4 vertice. [8] (W p ) = 3, where W p i a wheel with p 5 vertice. [8] (K m, n ) = m, where 2 m n and K m, n i a complete bipartite graph. IV. SPLIT DOMINATING NUMBER FOR SOME SPECIAL CLASS GRAPHS 4.1 K 4 Graph K 4 graph contain a vertex et V = {v 1, v 2, v 3, v 4 }. It ha the ubet namely D = {v 2, v 4 }. Every vertex of D i incident with < V - D >. If we remove D from V, then < V - D > will be diconnected. Therefore the plit domination number for K 4 graph i 2. i.e., ( G) P a g e

4 Figure 1: K 4 Graph. Figure 2: Thomen Graph. 4.2 Thomen Graph In Thomen graph the vertex et V = {v 1, v 2, v 3, v 4, v 5, v 6 } and it ha the ubet D = {v 4, v 5, v 6 } of vertex et. Which i incident with < V - D >. If we remove D from V, then D i diconnected. Therefore the plit domination number i greater than 2. i.e., ( G) Cycle Graph Cycle graph vertice V = {v 1, v 2, v 3, v 4, v 5 } and it ha a ubet D whoe vertice are {v 2, v 5 } and every vertex of D i incident with < V - D >. When D i removed from V, < V - D > i diconnected. Hence the plit domination number i greater than 2. i.e., ( G) Cube Graph Cube graph contain a vertex et V = {v 1, v 2, v 3, v 4, v 5, v 6, v 7, v 8 }. It ha the ubet namely D = {v 2, v 6, v 8 }. Every vertex of D i incident with < V - D >. If we remove D from V, then < V - D > will be diconnected. Therefore the plit domination number for cube graph i 3. i.e., ( G) 3. Figure 3: Cycle Graph. Figure 4: Cube Graph. Theorem 10. A plit dominating et D if there exit two vertice w 1, w 2 V - D uch that every w 1 - w 2 path contain a vertex D. Proof. Let D be any et plit dominating et of G. Then <V - D > ha atleat two vertice from different component. 748 P a g e

5 The vertice w 1 and w 2 are not adjacent, but they are connected by a path contain atleat one vertex in D. Hence the reult. Theorem 11. If D i a plit dominating et of G then <V - D > i a plit dominating et of G. Proof. Since D i a minimal plit dominating et of G. Let u conider a vertex v D. There exit u V - D uch that V doe not atify N(u) D = {v}. Therefore V - D i a dominating et of G and further it i a plit dominating et. Since < D > i diconnected. Hence V - D i alo a plit dominating et. Theorem 12. For any graph G then ir(g). (G) Proof. Let D be a maximum irredundant et of vertice of G. Clearly V - D i a dominating et of G. Then D ha atleat two vertice and every vertex in D i adjacent to ome vertex in V - D. Thi implie that V - D i a plit dominating et of G. Alo V - D i the maximum irredundant et. Therefore (G) Hence (G) ir(g). V D = ir(g). Theorem 13. If D i a P-N et of a 4-regular graph, then D i irredundant. Proof. Suppoe D i not irredundant. Then there exit w D with I(w, D) =. Neither w nor any neighbour of w i D - perfect. Therefore D i not a P-N et. Hence, D i irredundant. V. CONCLUSION In thi paper we dicued the relation between the plit domination number and irredundant number. Then here we invetigate the ufficient condition of the plit domination number for ome pecial graph. Alo we develop the contruction of graph uing the plit domination number and irredundant number. Domination and irredundant can tand together to facilitate the network communication proce, electrical network, etc. REFERENCES [1] S. Delbin Prema and C. Jayaekaran, Domination and Irredundant Number of 4-Regular Graph, International Journal of Innovative Reearch Explorer, 5(3), 2018, [2] S. Delbin Prema and C. Jayaekaran, The Detour irredundant number of a graph, International Journal of Pure and Applied Mathematic, 11 Aug 2017 (Accepted). [3] N. Mohanapriya, S. Vimal Kumar, J. Vernold Vivin and M.Venkatachalam, Domination in 4 - regular graph with girth 3, Proceeding of the National Academy of Science, India Section A: Phyical Science, 85(2), 2015, ] Peter Damachke, Irredundance number veru domination number, Dicrete Math., 89(1), 1991, P a g e

6 [5] E.J. Cockayne, S.M. Hedetniemi, S.T. Hedetniemi and C.M. Mynhardt, Irredundant and perfect neibourhood et in tree, Dicrete Math., 188, 1998, [6] Michael A. Henning, Irredundance perfect graph, Dicrete Math., 142, 1995, [7] V.R. Kulli and B. Janakiram, The plit domination number of a graph, Graph Theory Note of York, 32(3), 1997, [8] Stephen Hedetniemi, Fiona Knoll and Renu Lakar, Split Domination, Independence and Irredundance in Graph, arxiv: [math.co], 2, 2016, 1-9. [9] T. Tamizh Chelvam and S. Robinon Chellathurai, A note on plit Domination number of a graph, Journal of Dicrete Mathematical Science and Cryptography, 12(2), 2009, P a g e