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Clone Hop Domination Number of Some Total and Line Graphs International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 8, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com Clone Hop Domination Number of some Total and Line graphs G. Mahadevan Department of Mathematics, Gandhigram Rural Institute- Deemed to be University, Gandhigram. V. Vijayalakshmi Research scholar, Gandhigram Rural Institute – Deemed to be University, Gandhigram. Selvam Avadayappan Department of mathematics, VHNSN college, Virudhunagar. Abstract corresponding edges of G are adjacent in G. The graph L(G) Recently the concept of clone hop domination is called the line graph or the edge graph of G. number of a graph was introduced by G.Mahaevan et.al.,. A set S V is a hop dominating set of G, if for every vertex Definition 1.3 A set is a clone hop dominating set of vV S there exists u S such that d(u,v) = . A set G if S is a hop dominating set and <S> has at least one perfect S V is a clone hop dominating set of G, if S is a hop matching. The minimum cardinality of clone hop dominating set is called CHD-number of G and it is denoted by CHD(G). dominating set and <S> has atleast one perfect matching. The minimum cardinality taken over all clone hop dominating sets Illustration. is called clone hop domination number of G and it is denoted by CHD(G). In this paper we investigate this number for V7 V3 varities of total and line graphs. Keywords: Clone hop dominating set, hop dominating set. V4 AMS Subject Classification: 05C69 V1 V2 V5 The research work was supported by DSA (Departmental special assistance) under University Grants Commission- V6 V8 New Delhi . Figure 1.1 Here CHD-set = {v1,v2}, CHD(G) = 2. I. Introduction. II. Mani Result. In [2] the authors introduced the concept of Hop dominating set. A set of a graph G is a Hop 2.1. Clone hop domination number of some total graphs dominating set of G if for every vV S there exists The clone hop domination number for some total u S such that d(u, v) = 2. The minimum cardinality of a graphs can be found, and are given as follows. hop dominating set is called a hop domination number and it Theorem 2.1.1. For any Pn, n ≥ 3, is denoted by h (G) . A vertex u in a hop dominating set is n said to hop dominate an another vertex v if d(u, v) = 2. 4 2 if n 1(mod6); CHD(T(Pn)) = 6 Motivated by the above we introduced the new concept of n clone hop dominating set. A set is a clone hop 4 otherwise. 6 dominating set of G if S is a hop dominating set and <S> has atleast one perfect matching. The minimum cardinality of Proof. Let V(Pn) = {v1, v2,…...vn}, clone hop dominating set is called clone hop domination 2 3 n number of G and it is denoted by CHD(G). V(T(Pn)) = V(Pn) {v1 ,v2 ,.....vn1}. Definition 1.1 The total graph of G, denoted by T(G) is Let E(Pn) = { vivi1 :1 i n}, defined as the vertex set of T(G) is . Two vertices x, y in the vertex set of T(G) are adjacent in T(G) in case one E(T(Pn)) = E(Pn) of the following holds: (i) x, y are in V(G) and x is adjacent to i1 i2 i1 . y in G. (ii) x, y are in E(G) and x, y are adjacent in G. (iii) x is {vi vi1 :1 i n 2} {vivi :1 i n} in V(G), y is in E(G) and x, y are incident in G. Let S1 {v ,v ,v j ,vk : i 3(mod6), i j i j . Definition 1.2 Let G be a loopless graph. The vertex set of L(G) is in 1-1 correspondence with the edge set of G and two j 4(mod6),k 5(mod6),1 i, j,k n} vertices of L(G) are joined by an edge if and only if the Page 72 of 76 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 8, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com Now let S = Here the darkned vertices are denoted as the CHD set. Here S 1 if n 0,5(mod 6); 5 CHD(T(P5)) = 4 = 4, 1 n 6 S {vn ,vn1} if n 1(mod 6); 1 n1 n S {vn1 ,vn ,vn2 ,vn1} if n 2,3,4(mod 6). 7 CHD(T(P7)) = 4 2 = 8-2 = 6. Clearly 6 S1 if n 0,5(mod 6); Theorem 2.1.2. For any Cn , n ≥ 3, S 1 n S {vn ,vn1} if n 1(mod 6); 1 n1 n CHD(T(Cn)) = S {vn1,vn ,vn2 ,vn1} if n 2,3,4(mod 6). and S is a CHD set for T(Pn). Hence Proof. Let V(Cn) = {v1, v2, ..vn}, 2 3 1 V(T(Cn)) = V(Cn) {v ,v ,.....v }. CHD(T(Pn)) ≤ 1 2 n 1 j k Let S {vi ,v j ,vi ,v j : i 3(mod6), Let S be any CHD set for T(Pn). j 4(mod6),k 5(mod6),1 i, j,k n} i1 1 dT ( p ) (vi ,vi2 ) dT ( p ) (vi ,vi2 ) 2. S if n 0,5(mod6); n n Now let S = S 1 {v ,vn } if n 1(mod6); i2 n n1 dT ( p ) (vi ,vi1 ) 2. 1 n1 n n S {vn1,vn ,vn2 ,vn1} if n 2,3,4(mod6). i1 i1 i1 i3 d (v ,v ) d (v ,v ) 2 The set S and S1 is same as the above theorem. T ( pn ) i i2 T ( pn ) i i2 Thus the proof is follows from the above theorem. d (vi1,v ) d (vi1,v ) 2. Illustration. T ( pn ) i i1 T ( pn ) i i2 V1 Thus 2 1 V1 V5 S1 if n 0,5(mod 6); V5 V2 S 1 n S {vn ,vn1} if n 1(mod 6); 1 n1 n 5 3 V4 V2 S {vn1,vn ,vn2 ,vn1} if n 2,3,4(mod 6). Hence V4 4 V3 V3 Figure 2.2 CHD(T(Pn)) ≥ Here the darkned vertices are denoted as the CHD set. Here n 4 5 2 if n 1(mod6); 6 CHD(T(C5)) = 4 =4. and the result follows. 6n 4 otherwise. Observation 2.1.3. 6 CHD number does not exist for the total Illustration. graph of complete graph. 2 3 4 5 V V2 V V 1 3 4 Theorem 2.1.4. For any K , m,n ≥ 2, m,n mn 3 if m,n odd; V1 V2 V3 V V5 4 CHD(T(Km,n)) = 2 3 4 5 6 7 mn 2 otherwise. V1 V2 V3 V 4 V5 V6 Proof. Let V(Km,n) = {v1, v2, …..vm, w1, w2, …..wn} and V V7 V1 V2 V3 V V5 6 j 4 V(T(Km,n)) = V(Km,n ) {vi :1 i m,1 j n}. Figure 2.1 Let S1 = {v ,w ,v j :1 i m,1 j n} . 1 1 i Page 73 of 76 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 8, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com 1 1 ≤ i ≤ n. S {v2} if m,n odd; Let S = 1 2 3 1 Let S = . Clearly S is a CHD set for T(W ) . S otherwise. {v0 ,v0 ,v0 ,v0 } 1,n Clearly Hence CHD(T(W1,n)) = 4 for all n ≥ 3. Illustration. 6 S V1 V6 V V 1 1 6 6 0 V5 V0 2 and S is a CHD set for T(Km,n). V1 V 2 Hence 0 V 5 V2 5 V0 V0 mn 3 if m,n odd; V 3 0 5 3 CHD(T(Km,n)) ≤ V4 V2 4 mn 2 otherwise. V0 4 V4 4 V3 d (v ,v ) d (w ,w ) 2 V3 T (Km,n ) i j T (Km,n ) i j for all 1 ≤ i,j ≤ m, i ≠ j. Figure 2.4 d (v ,w ) d (v ,w ) 1for all (Km,n ) i j T (Km,n ) i j Here the darkned vertices are denoted as the CHD set. Here 1 ≤ i ≤ m, 1 ≤ j ≤ n. CHD(T(W1,6)) = 4. d (v j ,vk ) d (v j ,v j ) 1for all T (Km,n ) i i T (Km,n ) i l 2.2. Clone hop domination number of some line graphs 1 ≤ j,k ≤ m, 1 ≤ i, l ≤ n, k ≠ j, ≠ i. The clone hop domination number for some line graphs can be found, and are given as follows. Thus S Hence Theorem 2.2.1. For any Pn, n ≥ 4, n 1 CHD(L(Pn)) = 2 6 CHD(T(Km,n)) ≥ Proof. Let V(Pn) = {v1, v2, …….vn}, Thus the result follows. E(Pn) = {e1, e2,……….en-1}. Illustration. Therefore V(L(Pn)) = {e1, e2,……….en-1}. 1 Let S = {ei,ei+1: i 3(mod6),1 i n 2 }. V1 V 2 V3 3 1 V1 V 1 2 V3 S1 if n 0,4,5(mod6); 2 Let S = V3 1 2 S {en2 ,en1} if n 1,2,3(mod6). V 1 V1 1 3 2 V3 V2 d (e ,e ) 2 for all 3 ≤ i ≤ n-3.
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