Eccentric domination in splitting graph of some graphs

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1 Advances in Theoretical and Applied Mathematics ISSN Volume 11, Number 2 (2016), pp Research India Publications Eccentric domination in splitting graph of some graphs M. Bhanumathi 1 and R. Niroja 2 1 Associate Professor, Govt. Arts College for Women, Pudukkottai , Tamilnadu, India. bhanu_ksp@yahoo.com 2 Research Scholar, Govt. Arts College for Women, Pudukkottai , Tamilnadu, India. nironeela@gmail.com Abstract A subset D of the vertex set V(G) of a graph G is said to be a dominating set if every vertex not in D is adjacent to at least one vertex in D. A dominating set D is said to be an eccentric dominating set if for every v V D, there exists at least one eccentric vertex of v in D. The minimum of the cardinalities of the eccentric dominating set of G is called the eccentric domination number γ ed (G) of G. For a graph G, let V'(G) = {v': v V(G)} be a copy of V(G). Then the splitting graph S p (G) of G is the graph with the vertex set V(G) V'(G) and edge set{uv, u'v, uv': uv E(G)}. In this paper, we initiate the study of eccentric domination in splitting graph of a graph. Keywords: Eccentric dominating set, Eccentric domination number, Splitting graph. Mathematics Subject Classification: 05C12, 05C INTRODUCTION Let G be a finite, simple, undirected (n, m) graph with vertex set V(G) and edge set E(G), V(G) = n, E(G) = m. For graph theoretic terminology refer to Harary [4], Buckly and Harary [1]. Janakiraman, Bhanumathi and Muthammai [5] introduced Eccentric domination in Graphs. Bhanumathi and Muthammai [2] studied bounds on Eccentric domination in Graphs. Splitting graphs were first studied by Sampathkumar and Walikar [8] and were further developed by Patil and Thangamari [7].

2 180 M. Bhanumathi and R. Niroja Janakiraman, Muthammai and Bhanumathi studied eccentricity properties of splitting graphs in [6]. Let G be a connected graph and v be a vertex of G. The eccentricity e(v) of v is the distance to a vertex farthest from v. Thus, e(v) = max{d(u, v): u V}. The radius r(g) is the minimum eccentricity of the vertices, whereas the diameter diam(g) = d(g) is the maximum eccentricity. For any connected graph G, r(g) diam(g) 2r(G). v is a central vertex if e(v) = r(g). The center C(G) is the set of all central vertices. For a vertex v, each vertex at a distance e(v) from v is an eccentric vertex of v. Eccentric set of a vertex v is defined as E(v) = {u V(G) / d(u, v) = e(v)}. A set D V is said to be a dominating set in G, if every vertex in V D is adjacent to some vertex in D. The minimum cardinality of a dominating set is called the domination number and is denoted by γ(g) [3, 10]. A set D V(G) is an eccentric dominating set if D is a dominating set of G and for every v V D, there exist at least one eccentric vertex of v in D. The minimum cardinality of an eccentric dominating set is called the eccentric domination number and is denoted by γ ed (G). If D is an eccentric dominating set, then every superset D D is also an eccentric dominating set. But D D is not necessarily an eccentric dominating set. An eccentric dominating set D is a minimal eccentric dominating set if no proper subset D D is an eccentric dominating set [5]. For a graph G, let V (G) = {v : v V(G)} be a copy of V(G). Then the splitting graph S p (G) of G is the graph with the vertex set V(G) V (G) and edge set {uv, u v, uv : uv E(G)}[7]. Clearly, (i) For any graph G, γ(g) γ ed (G).(ii)For any graph G, γ(s p (G)) γ ed (S p (G)). Theorem 1.1 [9]: For all n 2, 0 4 γ(s p (P n )) = 1, Theorem 1.2 [9]: For all n 3, 0 4 γ(s p (C n )) = 1, Theorem 1.3 [6]: Let G be any graph with at least three vertices and r(g) = 1.Then S p (G) is self-centered with radius 2 if and only if G has no pendant vertices. Corollary 1.3 [6]: If r(g) = 1 and G has pendant vertices, then S p (G) is bi-eccentric with radius two.

3 Eccentric domination in splitting graph of some graphs 181 Theorem 1.4 [6]: Let G be a self-centered graph with radius two. Then S p (G) is also self-centered with radius two if and only if for every pair of adjacent vertices u, v in G, N G (u) N G (v) φ. Corollary 1.4 [6]: Let G be a self-centered graph with radius two. If there exists a pair of adjacent vertices v i, v j in G with N G (v i ) N G (v j ) = φ, then, S p (G) is bi-eccentric with radius two. Theorem 1.5 [6]: r(s p (G)) = max{2, r(g)}. Theorem 1.6 [5]: If G is of diameter two γ ed (G) 1+δ(G). Corollary [5]: If G is of radius 2 and diameter three, then γ ed (G) min {n deg G u/2, (n + deg G u 1)/2)}, where the minimum is taken over all central vertices. Corollary [5]: If G is of radius two and diameter three and if G has a pendant vertex v of eccentricity three then γ ed (G) Δ(G)+1. Theorem 1.7 [5]: If G is of radius two with a unique central vertex u then γ ed (G) n deg(u). Theorem 1.8 [5]: If G is of radius greater than two, then γ ed (G) n Δ(G). 2. ECCENTRIC DOMINATION IN SPLITTING GRAPHS Let us find out the eccentric domination of S p (G) of the following graph G. D = {v 2, v 5 } is a dominating set of S p (G), γ(s p (G)) = 2. S = {v 2, v 5, v 5 ', v 6 } is an eccentric dominating set of S p (G), γ ed (S p (G)) = 4.

4 182 M. Bhanumathi and R. Niroja Observation: 2.1 For any (n, m) graph G, which has no isolated vertices, 2 γ ed (S p (G)) n.the bounds are sharp, γ ed (S p (G)) = 2 if and only if G =P 2 and γ ed (S p (G)) = n if and only if G C n, n is even or G K n. Theorem: 2.1 If S p (K n ) is a splitting graph of complete graph K n, then γ ed (S p (K n )) = n, n > 2. Proof: Let v 1, v 2,, v n be the vertices of the complete graph K n which are duplicated by the vertices v 1, v 2,, v n respectively in S p (K n ). Then the resulting graph S p (K n ) will have 2n vertices. When G = S p (K n ), G is a graph with radius 2 and diameter 2. D = {v i, v i }, (i = 1, 2,, n) is a γ-set of G. Therefore, γ(g) = 2. S = {v 1, v 2,, v n }and {v 1, v 2,, v n } areeccentric dominating sets of G. Thus, γ ed (G) n.(1) S p (K n ) is self-centered of radius 2, v i has exactly only one eccentric vertex v i, i = 1, 2,, n. Hence either v i or v i must lie in a γ ed -set for i = 1, 2,, n. Therefore, γ ed (G) n. (2) From (1) and (2), γ ed (G) = n. Remark: 2.1 γ ed (S p (K 2 )) = 2. Theorem: 2.2 If S p (W n ) is a splitting graph of wheel graph W n = K 1 + C n, then γ ed (S p (W n )) = 3 for n> 3. Proof: Let v, v 1, v 2,, v n be the vertices of the wheel graph W n which are duplicated by the vertices v, v 1, v 2,, v n respectively in S p (W n ), where v is the central vertex of W n and v 1, v 2,, v n are the vertices of cycle C n. Then the resulting graph S p (W n ) will have 2(n+1) vertices. When G = S p (W n ), G is a graph with radius 2 and diameter 2.D = {v, v } is a γ-set of G. Therefore, γ(g) = 2. S = {v, v i, v i+1 } and {v, v i, v i+1 }, (i =1, 2,, n-1) are eccentric dominating sets of G.Thus, γ ed (G) 3. (1) S p (W n ) is self-centered of radius 2, v is an eccentric vertex of v, {v i, v i+1 } and {v i, v i+1 }are eccentric vertex sets of G. An eccentric vertex set must lie a subset of a γ ed - set of G.Also there exists no γ-set which is a γ ed -set for G. Therefore, γ ed (G) 3. (2) From (1) and (2), γ ed (G) = 3. Remark: 2.2γ ed (S p (W 3 )) = 4. Theorem: 2.3 If S p (F n ) is a splitting graph of Fan graph F n =K 1 + P n, thenγ ed (S p (F n )) = 3. Proof: Let v, v 1, v 2,, v n be the vertices of the fan graph F n which are duplicated by the vertices v, v 1, v 2,, v n respectively in S p (F n ), where v is the central vertex of F n and v 1, v 2,, v n are the vertices of path P n. Then the resulting graph S p (F n ) will have 2(n+1) vertices. When G = S p (F n ), G is a graph with radius 2 and diameter 2. D = {v,

5 Eccentric domination in splitting graph of some graphs 183 v } is a γ-set of G. Therefore, γ(g) = 2. S = {v, v i, v i+1 } and {v, v i, v i+1 }, (i =1, 2,, n 1) areeccentric dominating sets of G. Thus, γ ed (G) 3. (1) S p (F n ) is self-centered of radius 2, v is an eccentric vertex of v, {v i, v i+1 } and {v i, v i+1 }are eccentric vertex sets of G. An eccentric vertex set must lie a subset of a γ ed - set of G. Also there exists no γ-set which is a γ ed -set for G. Therefore, γ ed (G) 3.(2) From (1) and (2), γ ed (G) =3. Theorem: 2.4 If S p (K 1, n ), n 1 is a splitting graph of star graph K 1, n, then 2 1. γ ed (S p (K 1, n )) = 3 1. Proof: Let v, v 1, v 2,, v n be the vertices of the star graph K 1, n which are duplicated by the vertices v, v 1, v 2,, v n respectivelyin S p (K 1, n ), where v is the central vertex of K 1, n. Then the resulting graph S p (K 1, n ) will have 2(n+1) vertices. When G = S p (K 1, n), G is a graph with radius 2 and diameter 3. D = {v, v } is a γ-set of G. Therefore, γ(g) = 2. Case (i): n = 1 S = {v, v 1 } and {v, v 1 } are minimum eccentric dominating sets of G. Thus, γ ed (G) = 2. Case (ii): n > 1 S = {v, v, v i } and {v, v, v i }, 1 i n are eccentric dominating sets of G. Thus, γ ed (G) 3. (1) v is an eccentric vertex of vand v i is an eccentric vertex of {v i, v }.Also there exists no γ-set which is aγ ed -set for G. Therefore, γ ed (G) 3. (2) From (1) and (2), γ ed (G) = 3. Theorem: 2.5 If S p (P n K 1 ) is a splitting graph of corona graph P n K 1, then γ ed (S p (P n K 1 )) = n + 2, n > 3. Proof: Let A = {v 1, v 2,, v n } be the set of vertices of P n and B = {u 1, u 2,, u n } be the set of pendent vertices attached at v 1, v 2,, v n respectively. Let u 1, u 2,, u n, v 1, v 2,, v n be the duplicate vertices of u 1, u 2,, u n, v 1, v 2,, v n respectivelyin S p (P n K 1 ). Then the resulting graph S p (P n K 1 ) will have 4n vertices. When G = S p (P n K 1 ), G is a graph with radius and diameter n + 1. D = {v 1, v 2,, v n } is the only γ-set of G. Therefore, γ(g) = n.u 1, u n, u 1, u n are the eccentric vertices of G. {w 1, w n }, where w 1 = u 1 andu 1, w n = u n andu n are eccentric vertex sets of G. An eccentric vertex set must lie a subset of a γ ed -set of G.S = {v 1, v 2,, v n, w 1, w n } is a γ ed -set of G. Therefore, γ ed (G) = n+2. Remark: 2.3γ ed (S p (P 2 K 1 )) = 4, γ ed (S p (P 3 K 1 )) = 6.

6 184 M. Bhanumathi and R. Niroja Theorem: 2.6If S p (C n K 1 )) is a splitting graph of corona graph C n K 1, then γ ed (S p (C n K 1 )) =. 2. Proof: Let A = {v 1, v 2,, v n } be the set of vertices of C n and B = {u 1, u 2,, u n } be the set of pendent vertices attached at v 1, v 2,, v n respectively. Let u 1, u 2,, u n, v 1, v 2,, v n be the duplicate vertices of u 1, u 2,, u n, v 1, v 2,, v n respectively in S p (C n K 1 ). Then the resulting graph S p (C n K 1 ) will have 4n vertices. Case (i): n is odd When G = S p (C n K 1 ). G is a graph with radius and diameter + 2.D = {v 1, v 2,, v n } is the only γ -set of G. Therefore, γ(g) = n. u 1, u 3,, u n, u 1, u 3,, u n are the eccentric vertices of G. W = {u 1, u 3,, u n } and{u 1, u 3,, u n }are eccentric vertex sets of G. An eccentric vertex set must lie a subset of a γ ed -set of G.S = {v 1, v 2,, v n } W is a γ ed -set of G. Therefore, γ ed (G) =. Case (ii): n is even When G = S p (C n K 1 ). G is a graph with radius and diameter +2. D = {v 1, v 2,, v n } is the only γ-set of G. Therefore, γ(g) = n.u 1, u 2,, u n, u 1, u 2,, u n are the eccentric vertices of G. W = {u 1, u 2,, u n } and{u 1, u 2,, u n } are eccentric vertex sets of G.An eccentric vertex set must lie a subset of a γ ed -set of G.S = {v 1, v 2,, v n } Wis a γ ed -set of G. Therefore, γ ed (G) = 2n. Remark: 2.4γ ed (S p (C 3 K 1 )) = 6. Theorem: 2.7 If S p (P n ) is a splitting graph of path P n, then n + 1 if n = 4k + 1, k 2. 2 n + 2 if n = 4k + 2, k 1. γ = 2 ed ( S p( Pn )) n + = 1 if n 4k 1, k 3. 2 n + 4 if n = 4k, k 2. 2 Proof: Let v 1, v 2,, v n be the vertices of the path P n which are duplicated by the vertices v 1, v 2,, v n respectively in S p (P n ). Then the resulting graph S p (P n ) will have 2n vertices. Case (i): n = 4k+1, k 2

7 Eccentric domination in splitting graph of some graphs 185 When G = S p (P n ), G is a graph with radius and diameter n 1. S = {v 1, v 2, v 4,, v 4k-3, v 4k, v 4k+1 }is a minimum eccentric dominating set of G. Thus, γ ed (G) = +1. Case (ii): n = 4k+2, k 1 When G = S p (P n ), G is a graph with radius and diameter n 1. S = {v 1, v 2, v 5,, v 4k- 7, v 4k-6, v 4k-3, v 4k-2, v 4k+1, v 4k+2 } is a minimum eccentric dominating set of G. Thus, γ ed (G) =. Case (iii): n = 4k 1, k 3 When G = S p (P n ), G is a graph with radius and diameter n 1. S = {v 1, v 2, v 5,, v 4k-7, v 4k-6, v 4k-5, v 4k-2, v 4k-1 } is a minimum eccentric dominating set of G. Thus, γ ed (G) = +1 Case (iv): n = 4k, k 2 When G = S p (P n ), G is a graph with radius and diameter n 1. S = {v 1, v 2, v 5,, v 4k- 6, v 4k-5, v 4k-4, v 4k-1, v 4k } is a minimum eccentric dominating set of G.Thus, γ ed (G) = Remark: 2.5γ ed (S p (P 2 )) = 2, γ ed (S p (P 3 )) = 3, γ ed (S p (P 4 )) = 4, γ ed (S p (P 5 )) = 5, γ ed (S p (P 7 )) = 5. Theorem: 2.8If S p (C n ) is a splitting graph of cycle C n, then 2n if n = 3k for k n = + if n 3k 1 for k 2. γ ed ( S p( Cn)) = 3 2n if n = 3k + 2 for k 3. 3 n if n = 2k for k 3. Proof: Let v 1, v 2,, v n be the vertices of the cycle C n which are duplicated by the vertices v 1, v 2,, v n respectively in S p (C n ). Then the resulting graph S p (C n ) will have 2n vertices. When n is odd: Case (i): n = 3k, k 3, k-odd. When G = S p (C n ), G is a graph with radius and diameter.s = {v 1, v 4, v 7,, v 3k- 8, v 3k-5, v 3k-2 } is a minimum eccentric dominating set of G.

8 186 M. Bhanumathi and R. Niroja Thus, γ ed (G) =. Case (ii): n = 3k+1, k 2, k-even. When G = S p (C n ), G is a graph with radius and diameter. S = {v 1, v 4, v 7,, v 3k- 6, v 3k-3, v 3k } is a minimum eccentric dominating set of G. Thus, γ ed (G) =. Case (iii): n = 3k+2, k 3, k-odd. When G = S p (C n ), G is a graph with radius and diameter. S = {v 1, v 4, v 7,, v 3k- 4, v 3k-1, v 3k+2 } is a minimum eccentric dominating set of G. Thus, γ ed (G) =. When n is even: Case (i): k is odd When G = S p (C n ), G is a graph with radius and diameter. S = {v 1, v 2, v 3,, v 2k-1, v 2k } is a minimum eccentric dominating set of G. Thus, γ ed (G) = n. Case (ii): k is even When G = S p (C n ), G is a graph with radius and diameter. S = {v 1, v 2, v 3,, v 2k-2, v 2k-1, v 2k }is a minimum eccentric dominating set of G. Thus, γ ed (G) =n. Remark: 2.6γ ed (S p (C 3 )) = 3, γ ed (S p (C 4 )) = 4, γ ed (S p (C 5 )) = 4. Theorem:2.9If S p (K m, n ) is a splitting graph of complete bipartite graph K m, n, then γ ed (S p (K m, n )) = 4, m, n 2 Proof: Let A = {v 1, v 2,, v m } and B = {u 1, u 2,, u n } be the set of vertices of K m, n. Let v 1, v 2,, v m, u 1, u 2,, u n be the duplicate vertices of v 1, v 2,, v m, u 1, u 2,, u n respectively in S p (K m, n ). Then the resulting graph S p (K m, n ) will have 2(m + n) vertices. When G = S p (K m, n ), G is a graph with radius 2 and diameter 3. D = {v i, u j }, ( i = 1, 2,, m), ( j = 1, 2,.., n) areγ-sets of G. Therefore, γ(g) = 2. S = {v i, v i, u j, u j }, ( i = 1, 2,, m), ( j = 1, 2,..., n) are minimum eccentric dominating sets of G.Thus, γ ed (G) = 4. Remark: 2.7(i) γ ed (S p (K 1, 1 )) = 2 (ii) If 1 m n, γ ed (S p (K m, n )) = 3. Theorem: 2.10 Let G be any graph with at least three vertices and r(g) = 1 with no pendant vertices, then γ ed (S p (G)) 1+δ(G).

9 Eccentric domination in splitting graph of some graphs 187 Proof: By Theorem 1.3, S p (G) is self-centered with radius two. Therefore, By Theorem 1.6, γ ed (S p (G)) 1+ δ(s p (G)) = 1+δ(G), since δ(s p (G)) = δ(g). Therefore, γ ed (S p (G)) 1+δ(G). Corollary: 2.10 If r(g) = 1 and G has pendant vertices then γ ed (S p (G)) min{2n deg G (u), n + deg G (u) 1/2}. Proof: By Corollary 1.3, S p (G) is bi-eccentric with radius two. If u is a central vertex of G with minimum degree then u is a central vertex of S p (G) and deg u in S p (G) = 2 deg u in G. Therefore, By Corollary 1.6.1, γ ed (S p (G)) min {2n 2deg G u/2, (2n + 2deg G u 1)/2}.γ ed (S p (G)) min {2n deg G (u), n + deg G (u) 1/2}, where the minimum is taken over all central vertices. Theorem: 2.11 Let G be a self-centered graph with radius two and for every pair of adjacent vertices u, v in G, N G (u) N G (v) φ. Then γ ed (S p (G)) 1+δ(G). Proof: By Theorem 1.4, S p (G) is also self-centered with radius two. Therefore, By Theorem 1.6, γ ed (S p (G)) 1+ δ(s p (G)) = 1+δ(G), since δ(s p (G)) = δ(g). Therefore, γ ed (S p (G)) 1+ δ(g). Corollary: Let G be a self-centered graph with radius two. If for every pair of adjacent vertices v i, v j in G, N G (v i ) N G (v j ) = φ then γ ed (S p (G)) min{2n deg G (v), n + deg G (v) 1/2}. Proof: By Corollary 1.4, S p (G) is bi-eccentric with radius two. If v is a central vertex of G with minimum degree then v is a central vertex of S p (G) and deg v in S p (G) = 2 deg v in G. Therefore, By Corollary 1.6.1, γ ed (S p (G)) min {2n 2deg G v/2, (2n + 2deg G v 1)/2}. γ ed (S p (G)) min{2n deg G (v), n + deg G (v) 1/2}, where the minimum is taken over all central vertices. Corollary: If G is a graph of radius two and diameter three and has a pendant vertex v of eccentricity three then γ ed (S p (G)) 2Δ(G)+1. Proof: If u is a vertex of maximum degree Δ(G) in G then u is a vertex of maximum degree Δ(S p (G)) in S p (G) and deg u in S p (G) = 2 deg u in G. If G is bi-eccentric with radius two then S p (G) is also bi-eccentric with radius two. Also, since G has a pendant vertex, S p (G) has a pendant vertex. Therefore, By Corollary 1.6.2, γ ed (S p (G)) Δ(S p (G))+1 = 2Δ(G)+1, since Δ(S p (G)) = 2Δ(G). Therefore, γ ed (S p (G)) 2Δ(G)+1. Theorem: 2.12 If G is a graph of radius two with a unique central vertex then γ ed (S p (G)) 2(n deg G (u)).

10 188 M. Bhanumathi and R. Niroja Proof: By Theorem 1.5, S p (G) is of radius two and has a unique central vertex. If u is a central vertex of G then u is a central vertex of S p (G) and deg u in S p (G) = 2 deg u in G. Therefore, By Theorem 1.7, γ ed (S p (G)) 2n 2deg G (u) 2(n deg G (u)). Theorem: 2.13 If G is a graph of radius greater than two, then γ ed (S p (G)) 2(n Δ(G)). Proof: By Theorem 1.5, S p (G) is of radius greater than two. If u is a central vertex of G then u is a central vertex of S p (G) and deg u in S p (G) = 2 deg u in G. Therefore, By Theorem 1.8, γ ed (S p (G)) 2n Δ(S p (G)) = 2n 2Δ(G) = 2(n Δ(G)), since Δ(S p (G) ) = 2Δ(G). Therefore, γ ed (S p (G)) 2(n Δ(G)). References: [1] Samir K. Vaidya 1 and Nirang J.Kothari 2, 2013, Domination Integrity of splitting Graph of path and cycle, ISRN Combinatorics, 2013, Article ID , pp.7. [2] Bhanumathi, M., Muthammai, S., 2012, Further Results on Eccentric domination in Graphs, International Journal of Engineering Science, Advanced Computing and Bio-Technology, 3(4), pp [3] Janakiraman, T.N., Bhanumathi, M., Muthammai, S., 2010, Eccentric domination in graphs, International Journal of Engineering science, Computing and Bio-Technology, 1(2), pp [4] Janakiraman, T.N., Muthammai, S., Bhanumathi, M., 2007, On Splitting Graphs, Ars combinatorial 82, pp [5] Teresa W. Haynes, Stephen T. Hedetniemi, Peter J. Slater, 1998, Fundamentals of Domination in graphs, Marcel Dekkar, New Yark. [6] Patil, H.P., Thangamari, S., 1996, Miscellaneous properties of splitting graphs and Related concepts, in Proceedings of the National workshop on Graph Theory and its Application, Manonmaniam Sundaranar University, Tirunelveli, February 21-27, pp [7] Sampathkumar, E., Walikar, H.B., , On the splitting graph of a graph, J. Karnatak Univ. Sci., 25 and 26 (combined), pp [8] Buckley, F., Harary, F., 1990, Distance in graphs, Addison-Wesley, Publishing company. [9] Cockayne, E.J., Hedetniemi, S.T., 1977, Towards a theory of domination in graphs.networks, 7:pp [10] Harary, F., 1972, Graph theory, Addition-Wesley Publishing Company Reading, Mass.