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 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 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 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) {v ,v ,.....v }. 1 2 n1
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
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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); S1 {v ,v ,vn1 ,vn } if n 2,3,4(mod 6). CHD(T(Cn)) = n1 n n2 n1 and S is a CHD set for T(Pn). Hence
n Proof. Let V(Cn) = {v1, v2, ..vn}, 4 2 if n 1(mod6); 2 3 1 6 V(T(Cn)) = V(Cn) {v ,v ,.....v }. CHD(T(Pn)) ≤ 1 2 n n 4 otherwise. Let S1 {v ,v ,v j ,vk : i 3(mod6), 6 i j i j
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. i1 i1 Illustration. dT ( p ) (vi ,vi1 ) dT ( p ) (vi ,vi2 ) 2. n n V 1 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 S {v ,v ,v ,v } if n 2,3,4(mod 6). V4 V2 n1 n n2 n1
Hence V4 4 V3 V3 Figure 2.2 CHD(T(Pn)) ≥ Here the darkned vertices are denoted as the CHD set. Here 5 CHD(T(C5)) = 4 =4. and the result follows. 6 Observation 2.1.3. 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(K ) = {v , v , …..v , w , w , …..w } and m,n 1 2 m 1 2 n V V V V5 V6 V7 1 2 3 V4 j 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
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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 3 mn 3 if m,n odd; V0 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). 1 V1 V1 3 2 V3 V2 d (e ,e ) 2 for all 3 ≤ i ≤ n-3. L(Pn ) i i2 W1 W2 W3 Clearly Figure 2.3 S1 if n 0,4,5(mod6); S Here the darkned vertices are denoted as the CHD set. Here 1 S {en2 ,en1} if n 1,2,3(mod6). CHD(T(K3,3)) = 9+3 =12. n 1 Theorem 2.1.5. For any W1,n, CHD(T(W1,n)) = 4 for all n ≥ 3. Thus CHD(L(Pn)) ≤ 2 6 Proof. Let V(W1,n) = {v0, v1, v2, …..vn} and V(T(W1,n)) = Let S be any CHD set for L(Pn). In Pn each edges ei is adjacent i i1 V(W1,n ){v0 :1 i n}{vi :1 i n,n 11} to ei±1. dT (W ) (v0 ,vi ) dT (W ) (v0 ,vi ) 1 for 1≤ i ≤ n. 1,n 1,n Therefore in L(Pn) each vertices ei is adjacent to ei±1.
d (v ,vi1) d (v ,v ) 2 for all Now again we get a new path with n 1 vertices. T (W1,n ) 0 i T (W1,n ) i i1 Each ei, ei+1 hop dominate ei-1, ei-2, ei+2, ei+3 and itself.
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Therefore Thus CHD(L(Cn)) ≥ S1 if n 0,4,5(mod6); S S1 {e ,e } if n 1,2,3(mod6). Thus the result follows. n2 n1 Illustration. n 1 e1 Thus CHD(L(Pn)) ≥ 2 6
e Hence the result follows. 5 e2
Illustration.
e1 e2 e3 e4 e4 e3 Figure 2.5 Figure 2.6 Here the darkned vertices are denoted as the CHD Here the darkned vertices are denoted as the CHD set. Here set. Here CHD(L(P5)) = 5 CHD(L(C5)) = 2 2 . Theorem 2.2.2. For any Cn, n ≥ 4, 6
n Theorem 2.2.3. For any Kn, n ≥ 4, CHD(L(Kn)) = . CHD(L(Cn)) = 2 6 Proof. Let V(Kn) = {v1, v2, …….vn},
Proof. Let V(Cn) = {v1, v2, …….vn}, E(Kn) ={e1,e2 ,....en ,en1,...... e n(n3) }. n 2 E(Cn) = {e1, e2,……….en}.
Therefore V(L(Cn)) = {e1, e2,……….en}. Therefore
1 Let S = {ei,ei+1: i 3(mod6),1 i n }. V(L(Kn)) = {e1,e2 ,....en ,en1,...... e n(n3) }. n 2 Let Let S = {e1, e2, en+1,en+2}. S1 if n 0,4,5(mod6); S = deg (e ) 2(n 2) for 1 L(Kn ) i S {en2 ,en1} if n 1,2,3(mod6). n(n 3) all 1 i n . d (e ,e ) 2 for all 1 ≤ i ≤ n. L(Cn ) i i2 2
Clearly d (e ,e ) 1if ei, ej incident at a single vertex in Kn. L(Kn ) i j S1 if n 0,4,5(mod6); d (e ,e ) 2 if ei, ej incident at a different vertices in S Thus L(Kn ) i j 1 S {en2 ,en1} if n 1,2,3(mod6). Kn.
n Clearly S is a CHD set for L(Kn). CHD(L(Cn)) ≤ 2 6 Thus CHD(L(Kn)) = 4.
Let S be any CHD set for L(Cn). In Cn each edges ei is Illustration.
e2 e4 adjacent to ei±1. e1 e3 e5
Therefore in L(Cn) each vertices ei is adjacent to ei±1. Now again we get a new cycle with n vertices.
Each ei, ei+1 hop dominate ei-1, ei-2, ei+2, ei+3 and itself.
e6 e8 e10 Therefore e7 e9 Figure 2.7 S Here the darkned vertices are denoted as the CHD set. Here
CHD(L(K5)) = 4.
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Theorem 2.2.4. For any Km,n, m,n ≥ 2, CHD(L(Km,n)) = 4. for 1 i n,n 1 j 2n if ei, ej Proof. Let V(Km,n) = {v1, v2, …….vm, w1, w2, …….wn} and incident at a single vertex in W1,n. d (e ,e ) 2 for L(W1,n ) i j E(Km,n)={e1,e2 ,.....en ,en1,....e2n ,......
1 i, j n,n 1 j 2n if ei, ej incident at a different e ,...... e } (m1)n1 mn . vertices in W1,n.
ThereforeV(L(Km,n))={e1,e2 ,.....en ,en1,...... Clearly S is a CHD set for W1,n.
Thus CHD(L(W1,n)) = 4 for all n ≥ 3. e2n ,...... e(m1)n1,...... emn } Illustration. Let S = {e1, e2, en+1,en+2}. e2 e3 e1 e4 d (e ,e ) 1for 1 i, j m,1 k n,i j. L(Km,n ) ink jnk d (e ,e ) 1for 1 i m,1 j,k n, j k. L(Km,n ) in j ink
e5 e8
d (e ,e ) 2for e6 e L(Km,n ) in j knl 7 1 i,k m,1 j,l n,i j k l. Figure 2.9 Here the darkned vertices are denoted as the CHD set. Clearly S is a CHD set for L(Km,n). Thus CHD(L(Km,n)) = 4. Illustration. Here CHD(L(W1,4)) = 4.
References e e e4 e5 e1 2 3 [1] S.K. Ayyaswamy and C. Natarajan, Hop domination in e10 e6 e7 e8 e9 graphs, Mathematicae Graph Theory.
[2] C. Natarajan and S.K. Ayyaswamy, Hop domination in e e e e14 e15 11 12 13 graphs II, Analele Stiintifice ale Universitatii Ovidius
e20 Constanta, Seria Mathematica, vol.23(2), pp. 187-199, e16 e17 e18 e19 2015.
e e e e e21 22 23 24 25 [3] T. Haynes, S. Hedetniemi and P. Slater, Fundamentals Figure 2.8 of domination in graphs (New York: Marcel Dekker), 1998. Here the darkned vertices are denoted as the CHD set. Here [4] J.Paulraj Joseph, G. Mahadevan and A. Selvam On CHD(L(K5,5)) = 4. complementary perfect domination number of a graph, Theorem 2.2.5. For any W1,n, n ≥ 3, Acta Ciencia Indica, pp. 846-854, 2006.
CHD(L(W1,n)) = 4. [5] D. Michalak, On Middle and Total graphs with Proof. Let V(W1,n) = {v0, v1, …….vn }and Coarseness Number Equal 1, Lecture Notes in Mathematics, Volume 1018: Graph Theory, Springer E(W1,n) = {e1,e2 ,.....en ,en1,....e2n}. Verlag, Berlin, pp. 139 – 150, 1983.
Therefore [6] P. Siva Kota Reddy, Kavita, S. Permi, B. Prashanth, A Note On Line Graphs, International J.Math. Combin. V(L(W1,n)) = {e1,e2 ,.....en ,en1,....e2n} vol.1, pp.119-12, 2011. Let S = {e1, e2, en+1,en+2}. [7] M. Behzad, A criterion for the planarity of a total graph, d (e ,e ) 1for n+1 i, j 2n. L(W1,n ) i j Proc. Cambridge Philos. Soc., vol. 6(3), pp. 679-681, 1967. d (e ,e ) 1for 1 i, j n, j i 1. L(W1,n ) i j
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