D.K.M.College for Women (Autonomous), Vellore , Tamilnadu, India.

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1 American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at ISSN (Print): , ISSN (Online): , ISSN (CD-ROM): AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research) Inverse Split and Inverse Non-Split Two-Out Degree Equitable Domination Number of the Line graphs of Some Special Classes of Graphs K. Ameenal Bibi 1, P.Rajakumari 1, P.G. and Research Department of Mathematics, D.K.M.College for Women (Autonomous), Vellore-63001, Tamilnadu, India. Abstract: A dominating set D of G is said to be an inverse two-out degree equitable dominating set if for any two vertices u, v D such that od D (u) od D (v) whereod D (u) = N(v) V D. The minimum cardinality of a minimal inverse two-out degree equitable dominating set is called the inverse twoout degree equitable domination number and it is denoted by γ oe (G). The purpose of this paper is to introduce the concept of inverse split (non-split) two out degree equitable domination numbers of the line graphs of S n, P n, C n, G n, H n, W n, andk 1,n respectively. Keywords: Two-out degree equitable domination number, Inverse two-out degree equitable number, Inverse Split (Non-Split) two-out degree equitable number, Line graph. I. Introduction Let G= (V,E) be a simple, finite, connected and undirected graph. A dominating set D of a graph G=(V,E) is a split (non-split) dominating set if < V D > is disconnected (connected). The split (non-split) domination number of G is the minimum cardinality of a minimal split (non-split) dominating set in G. Let D be the minimum dominating set in G. If a non-empty subset D of V D is a dominating set in G, then D is called an inverse dominating set with respect to D. Furthermore, if < V D > is disconnected (connected) then D is called an inverse split (non-split) dominating set. The inverse split (non-split) domination number of G is the minimum cardinality of a minimal inverse split (non-split) dominating set in G. V.R Kulliet all introduced the concept of split domination non-split domination and inverse domination in graphs. Let G= (V, E) be a simple, finite, undirected and connected graph. Any undefined term in this paper may be found in Haynes T.W. et al (1998).A non-empty sub set D V of a graph G is a dominating set of G if every vertex in V D is adjacent to some vertex in D. The domination number γ(g) is the minimum cardinality taken over all the minimal dominating set of G. Let D be the minimum dominating sets of G. If V D contains a dominating set D then D is called the inverse dominating set of G with respect to D. The inverse dominating number γ (G) is the minimum cardinality taken over all the minimal inverse dominating sets of G. A dominating set D V of a graph G is a split (non-split) dominating set if the induced sub graph < V D > is disconnected (connected). The split (non-split) domination number γ s (G) (γ ns (G)) is the minimum cardinality of a minimal split (non-split) dominating set of G. K. Ameenal Bibi and R. Selvakumar (008) introduced the concept of inverse split and non- split domination numbers in graphs. Let D be the minimum inverse dominating set of G with respect to D. Then D is called an inverse split (non-split) dominating set of G if the induced subgraph < V D > is disconnected(connected).the inverse split (non split) domination number is denoted by γ s (G) (γ (G) ) and it is the minimum cardinality ns taken over all the minimal inverse split (non-split) dominating sets of G. Let G=(V,E) be a simple graph. Let n and m denote the order and size of the graph G respectively. For any vertex v V, the open neighbourhood of v is the set N(v) = {u V, uv E}. Anitha.A, Arumugam. S and Chellai.M introduced the out degree as the outdegree of v with respect to D is denoted by od D (u) = N(v) V D. Ali Sahal and V.Mathad introduced the concept of two-out degree equitable domination in graphs. A dominating set D of G is said to two-out degree equitable dominating set if for any two vertices u, v D such that od D (u) od D (v).the minimum cardinality of a minimal two-out degree equitable dominating set is called two-out degree equitable domination number and is denoted by γ oe (G). The line graph L(G) of a graph G= (V,E) is the graph with vertex set E(G) in which two vertices are adjacent if and only if the corresponding edges in G are adjacent [14]. In this paper, we initiated the concept of inverse split and inverse non-split two-out degree equitable domination numbers in the line graphs of the graphs AIJRSTEM 18-40; 018, AIJRSTEM All Rights Reserved Page 5

2 namelys n, P n, C n, G n, H n, W n andk 1,n. Here x is the greatest integer not greater than x and x is the least integer not less than x. 1.1 Gear Graph [1] The gear graph is a wheel graph with vertices added between pair of vertices of the outer cycle. The gear graph G n has n + 1 vertices and 3n edges. Let V(G n ) = {v i : 1 i n} {u i : 1 i n} {v} and E(G n ) = {e i = v i u i, 1 i n} {e i = v i v, 1 i n} {e i = u i v i+1, 1 i n, subscripts modulo n} where v is an externalvertex adjacent to every other vertexv i for 1 i n. 1. Helm Graph [1] The helm graphh n is the graph obtained from a n-wheel graph by adjoining a pendent edge at each vertex of the cycle. The helm graph H n has n + 1 vertices and 3n edges and V(G n ) = {v} {v i : 1 i n} {u i : 1 i n}ande(g n ) = {e i = v i v i+1, 1 i n 1} {e i = v i v, 1 i n 1} {e i = v i u i, 1 i n 1} where v is an externalvertex adjacent to every other vertexv i for 1 i n. 1.3 Wheel Graph [1] The wheel graph W n on n + 1 vertices is defined as W n = C n + K 1 where C n is n-cycle. Let V(W n ) = {v i : 1 i n} {v} and E(W n ) = {e i = v i v i+1, 1 i n, subscripts modulo n} {e i = vv i, 1 i n where v is an external vertex adjacent to every other vertex. 1.4 SunletGraph[15] The Sunlet graph on n vertices is defined as V(S n ) = {v 1, v v n } {u 1, u u n } where v i s are the vertices of cycles taken in cyclic order and u i s are pendent edges. 1.5 Line Graph [14] A Line graph L(G) of a graph G is a graph on the set of edges of G such that in L(G) there is an edge uv if and only if the vertices u and v are adjacent in G. II.Main Results on γ soe (G): Inverse Split Two-Out Degree Equitable Domination Number of the Line graphs of Some Special Graphs In this section, we obtained the inverse split two-out degree equitable domination number of the line graphs of Sunlet, Path and Cycle graphs. Definition.1 Whenever D is a dominating set ofg, V D is also a dominating set of G. Let D' V D is dominating set of G. Then D is called an Inverse dominating set of G. The inverse dominating set D of a graph G is called an inverse split two-out degree equitable dominating set if the induced sub graph < V D > is disconnected and if for any two vertices u, v D such that od D (u) od D (v) whereod D (u) = N(v) V D. The minimum cardinality of a minimal inverse split two-out degree equitable dominating set is called an inverse split two-out degree equitable domination number of G and is denoted by γ (G). soe Theorem. For the Line graph of Sunlet graph S n, the inverse split two-out degree equitable domination number n is:γ soe [L(S n )] = for n 4. By the definition of line graph, V(L(S n )) = E(S n ) = {u i : (1 i n)} {v i : {e i : 1 i n 1} {v n }where v i and u i represents the edge e i and e i (1 i n) respectively. v i for 1 i n if L(S n) is even Let D = v i for 1 i n 1 if L(S n ) is odd be an inverse dominating set of L(S n )and V D = v i 1 {u i 1 i n} for 1 i n if L(S n ) is even v i 1 {u i 1 i n} for 1 i n+1 if L(S n ) is odd Now, v i D then od D (v i ) = N(v i ) V D If L(S n ) is even, for i = 1 od D (v ) = N(v ) V D = (v 1, v 3, u, u 3 ) (v 1, v 3 v n 1 ) (u 1, u, u n )} = (v 1, v 3, u, u 3 ) AIJRSTEM 18-40; 018, AIJRSTEM All Rights Reserved Page 6

3 For i =, od D (v 4 ) = N(v 4 ) V D = (v 3, v 5, u 4, u 5 ) {(v 1, v 3, v 5 v n 1 ) (u 1, u, u n )} = (v 3, v 5, u 4, u 5 ) For i = n, od D(v n ) = N(v n ) V D = (v 1, v n 1, u 1, u n ) {(v 1, v 3 v n 1 ) (u 1, u, u n )} = (v 1, v n 1, u 1, u n ) If L(S n ) is odd, for i = 1 od D (v ) = N(v ) V D = (v 1, v 3, u, u 3 ) {(v 1, v 3, v 5, v n ) (u 1, u, u n )} = (v 1, v 3, u, u 3 ) For i = n 1, od D(v n 1 ) = N(v n 1 ) V D = (v n, v n, u n 1, u n ) {(v 1, v 3 v n, v n 1, v n ) (u 1, u, u n )} = (v n, v n, u n 1, u n ) Then od D (u) od D (v) for any u, v D.SoD is the inverse two-out degree equitable dominating set and < V D > disconnected. Therefore D is the inverse split two-out degree equitable dominating set. Hence, γ soe [L(S n )] = n for n 4. Observation.3 For the Line graph of S n, γ soe [L(S n )] γ soe [L(S n )]. Theorem.4 For the Line graph of P n, the inverse split two-out degree equitable domination number is γ soe [L(P n )] = n -1 for n 5. The line graph of P n has n 1 vertices and n edges, since the degree of any vertex in L(P n ) is except the initial and terminal vertices. Here, D is the inverse two-out degree equitable dominating set. It is found that the induced sub graph < V D > is disconnected. Therefore Dis the minimum inverse split two-out degree equitable dominating set. Hence,γ soe [L(P n )] = n -1 for n 5. Observation.5 For the Line graph of P n, γ soe [L(P n )] > γ soe [L(P n )]. Theorem.6 For the Line graph of the Cycle C n, the inverse split two-out degree equitable domination number is γ soe [L(C n )] = n 3 for n 3. The line graph of C n has n vertices and n edges. Each vertex is of degree. Here, the induced sub graph < V D > is disconnected. Therefore Dis the minimum inverse split two-out degree equitable dominating set. Hence,γ soe [L(C n )] = n 3 for n 3. Observation.7 For the Line graph of C n, γ soe [L(C n )] γ soe [L(C n )]. AIJRSTEM 18-40; 018, AIJRSTEM All Rights Reserved Page 7

4 III. Main Results on γ nsoe (G): Inverse Non-Split Two-Out Degree Equitable Domination Numbers of the Line graphs of Some Special Graphs In this section, we obtained the inverse non-split two-out degree equitable domination number of the line graphs of Gear graph, Helm graph, Wheel graph and Star graph. Definition 3.1 Whenever Dis a dominating set ofg, V D is also a dominating set of G. Let D' V D is dominating set of G. Then D is called an Inverse dominating set of G. The inverse dominating set D of a graph G is called an inverse non-split two-out degree equitable dominating set if the induced sub graph < V D > is connected and if for any two vertices u, v D such that od D (u) od D (v) whereod D (u) = N(v) V D. The minimum cardinality of a minimal inverse non-split two-out degree equitable dominating set is called an inverse non-split two-out degree equitable domination number of G and is denoted by γ nsoe (G). Theorem 3. For the Line graph of Gear graph G n, the inverse non-split two -out degree equitable domination number is : γ nsoe [L(G n )] = n. Let L(G n ) be the line graph of G n of order 3n. By the definition of line graph, V(L(G n )) = E(G n ) in which the set of vertices of L(G n ), {e i : 1 i n} induces a clique of order n. Let {(e 1, e, e n ) (e 1, e, e n ) (e 1, e, e n } be the vertices in L(G n ). Let D = {e i : 1 i n} be the inverse dominating set of L(G n ) and V D = {(e i e i ): 1 i n} Now, e i D then od D (e i ) = N(e i ) V D for i = 1, e 1 D then od D (e 1 ) = N(e 1 ) V D = (e n ", e 1 ", e 1 ) {(e i e i ): 1 i n} = (e n ", e 1 ", e 1 ) If e i D then od D (e i ) = N(e i ) V D for i n = (e i 1 ", e i ", e i ) {(e i e i ): 1 i n} = (e i 1 ", e i ", e 1 ) Then od D (e i ) od D (e j ), for any e i, e j D. Therefore D is an inverse two-out degree equitable dominating set. So, γ oe [L(G n )] = n and < V D > is connected. Then D is the minimum inverse non-split two-out degree equitable dominating set. Hence, γ nsoe [L(G n )] = n. 3.3 Observation on L(G n ): (i)γ nsoe [L(G n )] = γ nsoe [L(G n )] = n (ii)γ nsoe [L(G n )] + γ nsoe [L(G n )] = n (iii)γ nsoe [L(G n )] γ nsoe [L(G n )] = n (iv)γ nsoe [L(G n (V D))] = n (v)γ nsoe [L(G n (V D))] = n (vi)γ nsoe [L(G n (V D))] + γ nsoe [L(G n (V D))] n (vii)γ nsoe [L(G n (V D))] γ nsoe [L(G n (V D))] n Theorem 3.4 For the Line graph of Helm graph H n, the inverse non- split two -out degree equitable domination number is : γ nsoe [L(H n )] = n Let L(H n ) be the line graph of H n of order 3n. Let {e i e i e i : 1 i n} be the vertex set of L(H n ). LetD = {e i ": 1 i n} be an inverse dominating set of L(H n ) and V D = {(e i e i ): 1 i n} Now, e i " D then od D (e i ") = N(e i ") V D for i = 1, e 1 " D then od D (e 1 ") = N(e 1 ") V D = (e n, e 1, e 1 ) {(e i e i ): 1 i n} = (e n, e 1, e 1 ) If e i " D then od D (e i ) = N(e i ") V D for i n = (e i 1, e i, e i ) {(e i e i ): 1 i n} = (e i 1, e i, e 1 ) AIJRSTEM 18-40; 018, AIJRSTEM All Rights Reserved Page 8

5 Then od D (e i ") od D (e j "), for any e i ", e j " D. Therefore D is an inverse two-out degree equitable dominating set. So, γ oe [L(H n )] = n and the induced sub graph< V D > is connected. Then D is a minimum inverse non-split two-out degree equitable dominating set. Hence, γ nsoe [L(H n )] = n. 3.5 Observation on L(H n ): (i)γ nsoe [L(H n )] = γ nsoe [L(H n )] = n (ii)γ nsoe [L(H n )] + γ nsoe [L(H n )] = n (iii)γ nsoe [L(H n )] γ nsoe [L(H n )] = n (iv)γ nsoe [L(H n (V D))] = n (v)γ nsoe [L(H n (V D))] = n (vi)γ nsoe [L(H n (V D))] + γ nsoe [L(H n (V D))] n (vii)γ nsoe [L(H n (V D))] γ nsoe [L(H n (V D))] n Theorem 3.6 For the Line graph of Wheel graph W n, the inverse non- split two -out degree equitable domination number is : γ nsoe [L(W n )] = n if n is even does not exist if n is odd for n > 5 Case (i) if n is even Let L(W n ) be the line graph of W n of order n.by the definition of the line graph, V(L(W n )) = {e i = v i v i+1, 1 i n, subscripts modulo n} {e i = vv i, 1 i n}. Also, V(L(W n )) = n Let D = {e i : 1 i n } be an inverse dominating set of L(W n) and V D = {(e i 1 : 1 i n ) (e i : 1 i n)} Now, e i D then od D (e i ) = N(e i ) V D for i = 1, e Dthen od D (e ) = N(e ) V D = {e 1, e 3, e, e 3 } {(e 1, e 3,. e n 1 ) (e 1, e,. e n } = {e 1, e 3, e, e 3 }. If i =, e 4 Dthen od D (e 4 ) = N(e 4 ) V D = {e 3, e 5, e 4, e 5 } {(e 1, e 3,. e n 1 ) (e 1, e,. e n } = {e 3, e 5, e 4, e 5 }. If i = n, e n Dthen od D (e n ) = N(e n ) V D = {e 1, e n 1, e n, e 1 } {(e 1, e 3,. e n 1 ) (e 1, e,. e n } = {e 1, e n 1, e n, e 1 }. Then od D (e i ) od D (e j ), for any e i, e j D. So D is an inverse two-out degree equitable dominating set. It is found that the induced sub graph< V D > is connected. Then D is the minimum inverse n non-split two-out degree equitable dominating set. Hence, γ nsoe [L(W n )] = if n is even. Case (i) if n is odd, with n > 5 Let D be an inverse dominating set of L(W n ), if for any two vertices u, v D such that od D (u) od D (v), Here, od D (u) = N(u) V D od D (v) = N(v) V D = n + 1 (i. e) od D (u) od D (v). This shows that L(W n ) is a graph with n 5 (if n is odd) is does not satisfy the two-out degree equitable domination condition. Observation 3.7 For the Line graph of W n, γ nsoe [L(W n )] = γ nsoe [L(W n )] = n AIJRSTEM 18-40; 018, AIJRSTEM All Rights Reserved Page 9

6 Theorem3.8 For the line graph of Star graph K 1,n, the inverse non- split two -out degree equitable domination number is : γ nsoe [L(K 1,n )] = for n > 4. The line graph of K 1,n is of order n and L(K 1,n ) is a complete graph. Let D = {v 3, v 4 } be an inverse dominating set of L(K 1,n ) and V D = {v 1, v, v 5, v 6. v n } Now, v 3 D then od D (v 3 ) = N(v 3 ) V D = {v 1, v, v 5 v n } {v 1, v, v 5, v 6 v n } = {v 1, v, v 5 v n } = n Similarly, od D (v 4 ) = n. Then od D (v 3 ) od D (v 4 ) for any v 3, v 4 D. SoD is an inverse two-out degree equitable dominating set and the induced sub graph < V D > is connected. Then D is the minimum inverse non-split two-out degree equitable dominating set. Hence, γ nsoe [L(K 1,n )] = for n > 4. Observation 3.9 For the Line graph of K 1,n, γ nsoe [L(K 1,n )] = γ nsoe [L(K 1,n )] IV. Conclusion In this paper, we introduced the concept of inverse split (non-split) two-out degree equitable domination number of the line graphs of some standard graphs. Also their relationships with other domination parameters are discussed. We further extended this study on the Line graphs of some more special classes of graphs. V. References [1] Ameenal Bibi, K. and Selvakumar, R (008). The Inverse split and non-split domination numbers in graphs. Proc. of the International Conference on Mathematics and Computer Science, ICMCS 008,Dept. of Mathematics, Loyola College, Chennai July 5-6. [] Ameenal Bibi, K. and Selvakumar, R (009). The Inverse strong non-split r-domination number of agraph. Proc. of the National Conference on Industrial Applications of Mathematics, NCMA 009, PG and Research Dept. of Mathematics, Sacred Heart College(Autonomous) Tirupattur, Vellore Dist., March [3] Anitha, S. Arumugam, and Mustapha Chellai Equitable Domination in Graphs Discrete Math. Algorithm.Appl. 03, , (011). [4] G. Chartrand and L. Lesniak, Graphs and Digraphs, Chapman and Hall. CRC, 4th edition, 005. [5] Cockayne, E.J. and Hedetniemi S.T. (1977). Towardsa theory of domination in graphs. Networks, 7. pp [6] Haynes, T.W., Hedetniemi S.T. and Slater P.J.(1998). Domination in Graphs : Advanced Topics,Marcel Dekker Inc. New York, U.S.A. [7] Haynes, T.W., Hedetniemi S.T. and Slater P.J.(1998). Fundamentals of domination in graphs,marcel Dekker Inc. New York, U.S.A. [8] Kulli, V.R. and Janakiram B. (1997). The split domination number of a graph. Graph Theory notesof New York. Newyork Academy of Sciences,XXXII. pp [9] Kulli, V.R. and Janakiram B. (000). The non-splitdomination number of a graph. The Journal of Pureand Applied Math. 31(5). pp [10] Kulli, V.R. and Sigarkanti S.C. (1991). Inverse domination in graphs. National Academy ScienceLetters, 15. [11] Mathavan Pillai, K., Mahesh M.S., Selvam.G, Non-split two-out degree equitable domination number in graphs, Journal of Chemical and Pharamaceutical Sciences, ISSN: [1] Mohammed Alatif, Puttaswamy Rangaiah, Nayaka S.R., Boundary domination of line and middle graph of wheel graph families. International journal of computer Applications( ) volume 134-No.5, January 016. [13] A.Sahal.A, and Mathad.V, Two-out degree equitable domination in graphs Transaction on combinatorics vol.no. 3 (013). [14] Thilagavathi and Vernold Vivin J, On Harmonious coloring of line graph of central graph of Paths,Applied Mathematical Sciences, Vol 3, 009, No.5, [15] Vernold Vivin J, Venkatachalam.M, On b-chromatic number of sunlet graph and wheel graph families, Journal of the Egyptian Mathematical society(015) 3, AIJRSTEM 18-40; 018, AIJRSTEM All Rights Reserved Page 10