Journal of Analysis and Comutation, Vol 8, No 1, (January-June 2012) : 1-8 ISSN : 0973-2861 J A C Serials Publications DOMINATION IN DEGREE SPLITTING GRAPHS B BASAVANAGOUD 1*, PRASHANT V PATIL 2 AND SUNILKUMAR M HOSAMANI 3 Abstract Let G = (V, E) be a grah with V = S 1 S 2, S t T where each S i is a set of vertices having at least two vertices and having the same degree and T = V S i The degree slitting grah of G is denoted by DS(G) is obtained from G by adding vertices w 1, w 2,, w t and joining w i to each vertex of S i (1 i t) Let the vertices and the edges of a grah G are called the elements of G In this aer, we study the variation in domination from the grah G to the degree slitting grah DS(G) Also we establish many bounds on (DS(G)) in terms of elements of G but not in terms of elements of DS(G) 2000 Mathematics Subject Classification: 05C69 Keywords: Degree slitting grah, domination number, domatic number 1 INTRODUCTION The grahs considered here are finite, undirected without loos or multile edges Let G = ( V, E) be a grah and the vertices and edges be called the elements of G Any undefined term in this aer may be found in Harary [2] Let G = ( V, E) be a grah A set D V is a dominating set of G if every vertex in V D is adjacent to some vertex in D The domination number (G) of G is the minimum cardinality of a minimal dominating set of G For an early survey on domination refer [1] In [4], R Ponaraj and S Somasundaram have initiated a study of degree slitting grah DS(G) of a (, q) grah which is stated as follows Let G = (V, E) be a grah with V = S 1 S 2, S t T where each S i is a set of vertices having at least two vertices and having the same degree and T = V S i The degree slitting grah of G is denoted by DS(G) is obtained from G by adding vertices w 1, w 2,, w t and joining w i to each vertex of S i (1 i t) 1,2,3 Deartment of Mathematics, Karnatak University, Dharwad-580 003 * Corresonding Author E-mail: bbasavanagoud@gmailcom
2 B BASAVANAGOUD, PRASHANT V PATIL AND SUNILKUMAR M HOSAMANI In this aer, we study the variation in domination from the grah G to the degree slitting grah DS(G) Also we establish, many bounds on γ(ds(g)) in terms of elements of G but not in terms of elements of DS(G) In Figure 1, a grah G and its degree slitting grah DS(G) are shown Figure 1: Here S 1 = {1, 7}, S 2 = {3, 4}, S 3 = {2, 5, 6} and T = φ 2 RESULTS First we calculate the domination number of DS(G) for some standard class of grahs Proosition 1: For any ath G = P, 3 then γ(ds(g)) = 2 Proosition 2: For any wheel G = W, 5 then γ(ds(g)) = 2 Proosition 3: If G is a regular grah, then γ(ds(g)) = 1
DOMINATION IN DEGREE SPLITTING GRAPHS 3 Proof: Let G be any regular grah Then DS(G) = G + K 1 and (DS(G)) = 1 Hence γ(ds(g)) = 1 The next result gives the uer bound for γ(ds(g)) Theorem 21: For any grah G, γ(ds(g)) w i T, where w i ; 1 i t and T are as defined in the definition of DS(G) Proof: Let G be any grah By the definition of DS(G), we have V(DS(G)) = S 1 S 2, S t T and w i be the set of vertices of all the corresonding sets of S i ; 1 i t in DS(G) To rove the above in equality we consider the following cases Case 1: Let T = φ Since each w i ; 1 i t is indeendent in DS(G) If T = φ then clearly w i will be a maximal indeendent set in DS(G) Since every maximal indeendent set is a minimal dominating set Therefore, γ(ds(g)) = w i Case 2: Let T φ Then there exist at least one vertex in G which is not resent in S i ; 1 i t Since G is an induced subgrah of DS(G), to dominate all the vertices of DS(G) we need at lest D w i T vertices Hence γ(ds(g)) D w i T Theorem 22: For any grah G, γ(ds(g)) 2 Proof: Let G be any grah of order To rove the result we consider the following cases Case 1: If T = φ, G has atmost S i 2 Hence wi 2 Therefore by Theorem 21, we get γ(ds(g)) w i 2 2 Case 2: If T = φ Then G has atmost S i 2 T Hence wi 2 T Therefore by Theorem 21, we have γ(ds(g)) w i + T T + T 2 = 2 2
4 B BASAVANAGOUD, PRASHANT V PATIL AND SUNILKUMAR M HOSAMANI In the next theorem, we characterize the grahs whose degree slitting grah have domination number equal to the half of the order of G Theorem 23: Let G be any nontrivial connected grah G of even order Then γ(ds(g)) = 2 if and only if T = φ and cardinality of each Si ; 1 i t is even Proof: Let G be any nontrivial connected grah G of even order with T = φ and each S i ; 1 i t is even Then by using Theorem 21 and Theorem 22, we get γ(ds(g)) = w i = 2 = ( is even) 2 Conversely, let G be any nontrivial connected grah with γ(ds(g)) = 2 and T φ Then by Theorem 21, γ(ds(g)) w i T and by Theorem 22, we get γ(ds(g)) 2 This imlies that V(G) is odd and a contradiction to our assumtion Hence G must be a grah of even order with T = φ and S i is even Theorem 24: Let G be any nontrivial tree T Then γ(t) γ(ds(t)) Proof: Let G be any nontrivial tree T Let D = {v 1, v 2,, v i }; 1 i be the minimum dominating set of T Then γ(t) = D Let S i be the set of all equal degree vertices in a tree T and H = V S i Therefore D = w i (H D) is a dominating set of DS(G) Hence (DS(T)) D = w i (H D) D = γ(t) Next theorem gives the relation between indeendence number of G and the domination number of DS(G) Theorem 25: For any grah G, γ(ds(g)) β 0 (G) Further equality holds for G = K
DOMINATION IN DEGREE SPLITTING GRAPHS 5 Proof: For any grah G, we have β 0 (G) Also by Theorem 22, γ(ds(g)) 2 Therefore γ(ds(g)) β (G) 0 Theorem A [3] For any grah G, q γ(g) Furthermore, γ(g) = q if and only if each comonent of G is a star Theorem 26: Let G = (, q) be any nontrivial connected grah with maximum degree, then q + 1 γ(ds(g)) (G) + w i Further, the lower bound is attained if and only if each comonent of G is a star and the uer bound is attained for G = K Proof: First consider the lower bound Let q = 1, then G is a tree Therefore the lower bound follows from the fact that for any nontrivial tree γ(ds(g)) 2 For equality, suose each comonent of a grah G is a star then the result can be easily verified Conversely, let q + 1 = γ(ds(g)) Then G has exactly q comonents with domination number equal to one Thus G is a forest Since q + 1 = γ(ds(g)), therefore by Theorem A, each comonent of a grah G is a star Now consider the uer bound Let v be a maximum degree vertex in G Then v = (G) Since G is a subgrah of DS(G) and each w i is indeendent in DS(G) Therefore γ(ds(g)) w i {v} (G) + w i Suose G = K, then the equality can be easily verified Proosition 4: Let G be any (, q) grah having a vertex of (G) = 1 Then G = H + K 1 and DS(G) has the following roerties (i) If H is a comlete grah, then γ(ds(g))sets are recisely {w 1 }, {v 1 }, {v 2 },, {v } (ii) If H is a block (which is not comlete) then there exist an unique γ(ds(g)) set {v x, w i }; 1 i t in DS(G) By observing all these results we raise one oen roblem which is stated as follows Oen Problem: Characterize the grahs for which γ(g) = γ(ds(g)) A artial solution to the above roblem is as follows Theorem 27: For any grah G, γ(g) = γ(ds(g)) if G is one of the following grahs (i) G = K (ii) G = K m,n (iii) bistar or tristar Proof: If G is any one of the grah mentioned in the statement of the theorem, then one can easily verify the result
6 B BASAVANAGOUD, PRASHANT V PATIL AND SUNILKUMAR M HOSAMANI 3 DOMATIC NUMBER OF DS(G) Let G be a grah A artition of its vertex set V(G) is called a domatic artition of G if each class of is a dominating set in G The maximum number of classes of a domatic artition of G is called the domatic number of G and it is denoted by d(g) The domatic number was introduced by Cockayne and Hedetniemi [1] Theorem 31: For any grah G, d(ds(g)) δ(g) + 2 Proof: For any grah G, d(g) δ(g) + 1, equality holds if and only if G = K If G = K then DS(G) = K + 1 Hence the theorem Theorem 32: For any grah G having vertices, d(ds(g)) + d(()) DS G + 3 Proof: Let G be any grah having vertices Then by Theorem 31, d(ds(g)) δ(g) + 2 and d(ds ()G ) δ ()G + 2 ()G + 2 Therefore, d(ds(g)) + d(ds ()G ) δ(g) + ()G ) + 4 1 + 4 = + 3 Theorem 33: Let G be any grah having vertices Then d(ds(g)) + d(ds ()G ) = + 3 if and only if G = K or K Proof: Let d(ds(g)) + d(ds ()G ) = + 3 We roceed by induction on Suose, contrary to the assertion that G K or following cases K has vertices and d(ds(g)) + d(ds ()G ) = + 3 We consider the Case 1: In G if has an unique vertex of δ(g) = 0 then δ(g) = δ(ds(g)) and d(ds(g)) = 1, and ()G has a vertex of degree 1, which is unique, then ()G = 1 Therefore, d(ds(g)) + d(ds ()G ) = δ(g) + 2 + δ ()G + 2 = 2 + ( 2) + 2 + 2 < + 3
DOMINATION IN DEGREE SPLITTING GRAPHS 7 Case 2: 0 < δ(g) < + 1 (31) 2 By assumtion, d(ds(g)) + d(ds ()G ) = + 3 Therefore, d(ds( ()G ) = + 3 d(ds(g)) = + 3 (δ(g) + 2) (By Theorem 31) = δ(g) + 1 (32) If all dominating sets in a maximum D artition of DS ()G have at least two vertices Then 2d(DS ()G ) 2( δ(g) + 1) (using 32) = + 3, a contradiction Hence some vertex v dominates G and {v w i } dominates DS ()G, Therefore deg (v) = w DS () G i, deg () v = 1 and deg G (v) = 0 Hence δ(g) = 0 = δ(ds(g)), contradiction to (31) If δ(ds(g)) = δ(g) = 0, aly Case 1, to DS(G) otherwise aly Case 2, to DS(G) Theorem 34: For any grah G with vertices, d(ds(g)) + γ(ds(g)) + 2, equality holds if and only if G = K Proof: If G = K, trivially d(ds(g)) + γ(ds(g)) + 2 Therefore, let G have vertices and G K From Theorem 26, we have γ(d S (G)) (G) + w i, hence γ(ds(g)) δ(g) + w i We claim that this inequality, d(ds(g)) δ(g) + 2, is strict Suose not then, γ(ds(g)) = δ(g) G = δ(ds(g)) + 2 Using the fact that d(ds(g)) + 1 γ (()) DS G, d(ds(g)) + 1 d (()) DS G2 + ie, d(ds(g)) ( d(ds(g)) + 2) + 1 from which on solving this equation we get, d(ds(g)) = + 1 and d(ds(g)) = 1
8 B BASAVANAGOUD, PRASHANT V PATIL AND SUNILKUMAR M HOSAMANI In the revious case G = K while d(g) = 1 imlies G has an isolated vertex v Since G v K 1, γ(ds(g)) > 1 In this case (G) = 0, w i = 1 and T = {v} Hence the inequality, γ(ds(g)) δ(g) + w i is strict Thus one of the inequalities γ(ds(g)) δ(g) + w i or d(ds(g)) d(g) + 2 is strict Hence d(ds(g)) + γ(ds(g)) < + 2 as asserted Nordhaus-Gaddum tye results for DS(G) Theorem 35: Let G be any grah having vertices, then (i) γ(ds(g)) + γ(ds ()G ) (ii) γ(ds(g)) γ(ds ()G ) 2 2 2 ACKNOWLEDGEMENT This research was suorted by UGC-SAP DRS-II New Delhi, India: for 2010-2015 REFERENCES [1] E J Cockayne and S T Hedetniemi, Towards a Theory of Domination in Grahs, Networks, 7(1977), 247-261 [2] F Harary, Grah Theory, Addison-Wesley, Reading, Mass, (1969) [3] TW Haynes, ST Hedetniemi and PJ Slater, Fundamentals of Domination in Grahs, Marcel Dekker, Inc, New York (1998) [4] P Ponaraj and S Somasundaram, On the Degree Slitting Grah of a Grah, Nat Acad Sci Letters, 27 No 7 and 8 (2004), 275-278 [5] B Zelinka, Domatic Numbers and Degrees of Vertices of a Grah, Math Slovaca, 33(1983), 145-147