Intersection Graph on Non-Split Majority Dominating Graph

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1 Malaya J. Mat. S()(015) Intersection Grah on Non-Slit Majority Dominating Grah J. Joseline Manora a and S. Veeramanikandan b, a,b PG and Research Deartment of Mathematics, T.B.M.L College, Porayar, , India. Abstract Let G = (V,E) be a grah. The Minimal Non-Slit Majority Dominating Grah MNMD(G) of a grah G is the intersection grah defined on the family of all minimal non-slit majority dominating sets of vertices in G and the Common Minimal Non-Slit Majority Dominatng Grah CMNMD(G) of a grah G is the grah having the same vertex set as G with two vertices adjacent in CMNMD(G) if and only if there exists a minimal non-slit majority dominating set of G containing them. In this article, We have introduced these two concets and investigated some roerties of these grahs. Some characterization theorems are established and bounds of MNMD(G) and CMNMD(G) are also determined. Keywords: Minimal Non-Slit Majority Dominating Grah MNMD(G), Common Minimal Non-Slit Majority Dominating Grah CMNMD(G) 010 MSC: 05C0, 05C69. c 01 MJM. All rights reserved. 1 Introduction Unless mentioned otherwise, all grahs considered in this article are finite, undirected and simle. Any undefined term in this aer may be found in Haynes [1]. Let G = (V, E) be a grah with V(G) = and E(G) = q. A subset D of V(G) is a dominating set[1] of G if every vertex in V-D is adjacent to some vertex in D. A dominating set D of G is minimal if no roer subset of D is a dominating set. The domination number γ(g) of G is the minimum cardinality of a minimal dominating set in G. A subset D of V(G) is a majority dominating set [4] of G if atleast half of the vertices of G are either in D or are adjacent to the vertices of D. A majority dominating set D of G is minimal if no roer subset of D is majority dominating set. The majority domination number γ M (G) of G is the minimum cardinality of a minimal majority dominating set in G. A vertex v of G is said to be majority dominating vertex if d (v) This arameter was defined by Swaminathan and Joseline Manora. A Majority Dominating set D is said to be Non-Slit Majority Dominating set [5] if the induced subgrah V D is connected. A Non-Slit Majority dominating set D is minimal if D-{v} is not a non-slit majority dominating set of G for every v G. The minimum cardinality of minimal non-slit majority dominating set of G is called Non-Slit majority domination number and it is denoted by γ NSM (G). Kulli and Janakiram introduced the concet of Minimal Dominating Grah MD(G) [] and Common Minimal Dominating Grah CD(G) [3] for a grah G. Let G = (V, E) be the grah and F = {D 1, D, D 3,...D n } be the artition of all minimal dominating sets of G. Then the intersection grah Ω(F) is defined on the family of all minimal dominating sets of G. The minimal dominating grah MD(G) is the intersection grah where two vertices of MD(G) are adjacent if and Corresonding author. address: joseline manora@yahoo.co.in (J. Joseline Manora), mathveeramani@gmail.com(s. Veeramanikandan).

2 J. Joseline Manora & S. Veeramanikandan / Minimal Non-Slit Majority Dominating Grah of a Grah 477 only if there exists atleast an element common between the two dominating sets of G. The Common Minimal Dominating Grah CD(G) is the intersection grah having the vertex set as the vertex set of G with two vertices of CD(G) are adjacent if and only if there exists a minimal dominating set containing them. This concet is extended to the concet of minimal majority dominating sets in G. The Minimal Majority Dominating Grah of a grah G MMD(G)[7]is the intersection grah defined on the family of all minimal majority dominating sets of G where two vertices of MMD(G) are adjacent if and only if there exists atleast an element between the minimal majority dominating sets of G and CMMD(G) is the intersection grah with two vertices of CMMD(G) are adjacent if and only if there exists a minimal majority dominating set containing them. Let F ={D 1, D,..., D n }be the family of all minimal non-slit majority dominating sets of G. Then the intersection grah Ω(F) defined on the family of all minimal non-slit majority dominating sets of vertices in G. Minimal Non-Slit Majority Dominating Grah of a Grah MNMD(G) In this section, We define the minimal non-slit majority dominating grah MNMD(G) of a grah G. We have structured MNMD(G) for some standard grahs. Definition.1. The Minimal Non-Slit Majority Dominating Grah MNMD(G) of a grah G is the intersection grah defined on the family of all minimal non-slit majority dominating sets of vertices in G. Two vertices of MNMD(G) are adjacent if and only if there is a common element between the minimal non-slit majority dominating sets of G. Examle.1. For a grah G = K,, MNMD(G) is totally disconnected as every vertex of G is a minimal non-slit majority dominating set of G. 3 Main Results on MNMD(G) Theorem 3.1. For any connected grah G, MNMD(G) is connected if and only if D has no majority dominating vertex, where D is minimal non-slit majority dominating set of G. Proof. Let G be any connected grah and D be minimal non-slit majority dominating set of G. First, Assume that MNMD(G) is connected and v is a majority dominating vertex. Consider the following two cases: case1: If v is a cutvertex, D = {v} is a majority dominating set of G but D is not a non-slit majority dominating set of G as V D is disconnected. Therefore, D has no such majority dominating vertex v and MNMD(G) is connected. case: If v is not a cutvertex, D = {v} is a non-slit majority dominating set of G. Then there exists no vertex of MNMD(G) containing v. This imlies that no edge is incident on the vertex v and so MNMD(G) is disconnected, a contradiction. Therefore, MNMD(G) is connected.

3 478 J. Joseline Manora & S. Veeramanikandan / Intersection Grah on Non-Slit Majority Dominating Grah Theorem 3.. For any connected grah G, MNMD(G) is disconnected with atleast an isolate if and only if G has a majority dominating vertex which is not a cut vertex. Proof. Suose G is connected and MNMD(G) is disconnected with atleast an isolate v. Then D = {v} is a non-slit majority dominating set of G. Therefore, v is a majority dominating vertex which is not a cut vertex. Suose G is a cut vertex, V D is disconnected, which is absurd. The converse art is obvious. Proosition 3.1. For any tree T, δ(g) < 1, MNMD(T) is a connected grah. Proof. Let T be any tree. Suose δ (T) -1. Then γnsm(t) = 1 and MNMD(T) is disconnected. Consider δ (T) < 1. Let F be the family of all minimal non-slit majority dominating sets of G. Since D has no majority dominating vertex, by theorem [3.1], MNMD(T) is a connected grah. Theorem 3.3. For any tree T and >4, MNMD(T) is a comlete grah if every non-endant vertex is adjacent to atleast a endant. Proof. Let T be any tree and > 4. Suose every non-endant vertex v is adjacent to atleast one endant. Then v is a cut vertex and no minimal non-slit majority dominating set of G contains these non-endants. If any vertex v of T is a majority dominating vertex,d={v} is not a non-slit majority dominating vertex as V D is disconnected. Therefore only endants of T constitute non-slit majority dominating sets. Let m be the number of endants of T. Suose v is an element of D. Then every indeendent majority dominating set containing only endants of T including v are minimal non-slit majority dominating sets of G. Therefore any air of vertices of MNMD(T) are adjacent. Thus MNMD(T) is comlete. Corollary 3.1. If a grah G is star or double star, MNMD(G) is comlete. Theorem 3.4. For any grah G, MNMD(G) is totally disconnected if and only if every vertex of G is a majority dominating vertex which is not a cut vertex. Proof. Suose MNMD(G) is totally disconnected. This shows that all minimal non-slit majority dominating sets are of cardinality one. In this case,γ NS M(G) = 1 and every vertex is a majority dominating vertex which is not a cut vertex and hence the result. The converse art is obvious. Theorem 3.5. If G is the join grah of the two grah G 1 and G, MNMD(G) is a disconnected grah. Proof. Let G 1 and G be two grahs of different orders. Suose V(G 1 ) < V(G ) and G = G 1 + G is the join grah. Then every vertex of G 1 is a majority dominating vertex. Since no vertex of G is a cut vertex, we have atleast V(G 1 ) minimal non-slit majority domingating sets of G. Therefore MNMD(G) is disconnected with atleast V(G 1 ) isolates. Proosition 3.. Let G be a r-regular grah. MNMD(G) is totally disconnected if r -1 and MNMD(G) is connected if r< 1. Proof. Let G be a r-regular grah and D be minimal non-slit majority dominating set of G. Case 1: Suose r < 1. Then every vertex v of G is a majority dominating vertex and V D is connected. Therefore{v} is a minimal non-slit majority dominating set of G. Thus MNMD(G) is totally disconnected. Case : If r< 1, G has no majority dominating vertex and so D. By theorem [3.1], MNMD(G) is connected. Theorem 3.6. For any cycle C, MNMD(C ) is a regular grah with r = ( 3 ). Proof. Consider a cycle C with 6 and D be minimal non-slit majoriy dominating set of G. Each vertex of C is a majority dominating vertex which is not a cut vertex. This imlies that MNMD(G) is totally disconnected. Suose G = C with > 7. By result [5], γ NSM (C )=. Since D is connected and C is - regular, every minimal non-slit majority dominating set of G is minimum. Suose D 1 = {v 1, v, v 3,...v } is a non-slit majority dominating set of G then D = {v, v 3, v 4,...v 1} and D 3 = {v 3, v 4, v 5,...v } and so on. For two sets D 1 and D, there are -3 vertices in common and the same argument holds for any D i and D i+1. This imlies that ( 3 ) edges are exactly incident on each vertex v of C. Therefore C is regular.

4 J. Joseline Manora & S. Veeramanikandan / Minimal Non-Slit Majority Dominating Grah of a Grah 479 Theorem 3.7. For any cycle C, MNMD(G) = C if and only if = 7 and = 8. Proof. Assume that MNMD(C ) = C which is true for >8.By theorem [3.6], MNMD(C is ( ) 3 regular. For = 9, MNMD(G) is 4-regular and r >.So, When > 9, MNMD(G) is ( ) 3 -regular. Also, For 3 6, MNMD(G) is totally disconnected. Therefore our assumtion is wrong. Hence MNMD(C ) = C if = 7 and =8. The converse art is trivial. Proosition 3.3. For any grah G,1 γ NSM (MNMD(G)) 1. The bounds are shar. Proof. Suose a grah G with every non-endant vertex is adjacent to atleast one endant. Then MNMD(G) is comlete and γ NSM (MNMD(G)) = 1. Suose G is comlete biartite grah K m,n where m < n. Then there exists m majority dominating vertices which are not cut vertices of G. The resultant grah MNMD(G) contains exactly m isolates and a comonent. In this case γ NSM (MNMD(G)) = m < -1.If every vertex of G is a majority dominating vertex, MNMD(G) is totally disconnected and γ NSM (MNMD(G)) = 1. The bounds are shar for G being K 1, 1 and K. Proosition 3.4. For any grah G, γ NSM (G)+γ NSM (MNMD(G)). Proof. Case1: Suose γ NSM (G) = 1. Then by theorem [3.],MNMD(G) is disconnected with atleast an isolate. Therefore γ NSM (MNMD(G)) 1.Thus γ NSM (G) + γ NSM (MNMD(G)). Case: Suose γ NSM (G) = 1. Then G being K, P and K K1.Then γ NSM (G) + γ NSM (MNMD(G)) =. Thus γ NSM (G) + γ NSM (MNMD(G)). Proosition 3.5. If γ NSM (MNMD(G)) = 1 then γ M (MNMD(G)) + γ NSM (MNMD(G)) =. Proof. Suose γ NSM (MNMD(G)) = 1. Since γ M (G) γ NSM (G), γ M (G) = 1. Hence the result. 4 Common Minimal Non-Slit Majority Dominating Grah CMNMD(G) In this section, we introduce Common Minimal Non-Slit Majority Dominating Grah CMNMD(G) of a grah G. We have established the structures of CMNMD(G) for some standard grahs and also bounds of CMNMD(G). Definition 4.. The Common Minimal Non-Slit Majority Dominating Grah CMNMD(G) of a grah G is the grah having the same vertex set as G with two vertices adjacent in CMNMD(G) if and only if there exists a minimal non-slit majority dominating set in G containing them. Examle 4.. Let G = C be a grah. Then CMNMD(G) is regular with arallel edges. 5 Results On CMNMD(G) Proosition 5.6. If G has a majority dominating vertex then CMNMD(G) is disconnected with atleast an isolate. Proof. Let v be a majority dominating vertex of G. If v is a cut vertex, no minimal non-slit majority dominating set contains v. Therefore there exists no edge incident on v in CMNMD(G) imlying CMNMD(G) is disconnected with atleast an isolate. Suose v is not a cut vertex. Then D = {v} is a minimal non-slit majority dominating set of G. Again, CMNMD(G) is disconnected since γ NSM (G) = 1.

5 480 J. Joseline Manora & S. Veeramanikandan / Intersection Grah on Non-Slit Majority Dominating Grah Theorem 5.8. For any connected grah G, CMNMD(G) is connected if and only if G has no majority dominating vertex. Proof. Suose G and CMNMD(G) are connected.assume that G has majority dominating vertex v. If v is a cut vertex, no non-slit majority dominating set of G contains v as D v is disconnected. Therefore in CMNMD(G) no edge is incident on v imlying CMNMD(G) is disconnected, a contradiction. Similarly, If v is not a cut vertex then D ={v} is a minimal non-slit majority dominating set of G. Again there is no edge incident on v in CMND(G) imlying CMND(G) is disconnected, a contradiction. Therefore G has no majority dominating vertex. The converse art is obvious. Theorem 5.9. For any grah G, γ M (CMNMD(G) = if and only ifd (v) 1 v G. Proof. The roof is obvious. Proosition 5.7. For any connected grah G, MNMD(G) = CMNMD(G) = K if and only if every vertex of G is a majority dominating vertex. Proof. The roof is trivial. References [1] Haynes.T.W, Hedetniemi.S.T, Petet J.Slater, Fundamentals of Domination in Grahs,Marcel Dekker, New York, [] Kulli.V.R, Janakiram.B, The Minimal Dominating Grah, Grah Theory Notes of New York,XXXVIII, 1-15, [3] Kulli.V.R, Janaikiram.B, The Common Minimal Dominating Grah, Indian J.ure al.math.,7(): , February [4] Swaminathan.V, Joseline Manora.J, Majority Dominating Sets in Grahs, Jamal Academic Research Journal, vol.3, No., 75-8, 006. [5] J. Joseline Manora and S.Veeramanikandan, The Non - Slit Majority Domination Number of a Grah, Proceedings of ICOMAC Secial Issue of Jamal Academic Research Journal, [6] J. Joseline Manora and S.Veeramanikandan, The Non - Slit Majority Dominating set of a Grah, Malaya Journal of Mathematik, Secial issue 1, 015, [7] J.Joseline Manora and S.Veeramanikandan, The Minimal Majority Dominating Grah of a Grah, Journal of Grah Labelling, acceted. Received: August 06, 015; Acceted: Setember 8, 015 UNIVERSITY PRESS Website: htt://