Lower bounds on the minus domination and k-subdomination numbers

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1 Theoretical Computer Science 96 (003) Lower bounds on the minus domination and k-subdomination numbers Liying Kang a;, Hong Qiao b, Erfang Shan a, Dingzhu Du c a Department of Mathematics, Shanghai University, Shanghai 00436, China b Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China c Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455, USA Abstract A three-valued function f dened on the vertex set of a graph G =(V; E), f V { 1; 0; 1} is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v V, f(n[v]) 1, where N[v] consists of v and all vertices adjacent to v. The weight of a minus function is f(v )= v V f(v). The minus domination number of a graph G, denoted by (G), equals the minimum weight of a minus dominating function of G. In this paper, sharp lower bounds on minus domination of a bipartite graph are given. Thus, we prove a conjecture proposed by Dunbar et al. (Discrete Math. 199 (1999) 35), and we give a lower bound on ks (G) of a graph G. c 00 Elsevier Science B.V. All rights reserved. Keywords Domination number; Minus domination; k-subdomination 1. Introduction For a graph G =(V; E) with vertex set V and edge set E, the open neighborhood of v V is N (v)={u V uv E} and the closed neighborhood of v is N[v]={v} N(v). For a set S of vertices, we dene the open neighborhood N(S)= v S N(v), and the closed neighborhood N [S]=N(S) S. Adominating set S for a graph G =(V; E) is a subset of the vertex set V such that every vertex v V is either in S or adjacent to a vertex in S. The domination number of G, (G), equals the minimum cardinality of a dominating set. This work was supported by NSFC. Corresponding author. address kangliying39@hotmail.com (L. Kang) /03/$ - see front matter c 00 Elsevier Science B.V. All rights reserved. PII S (0)

2 90 L. Kang et al. / Theoretical Computer Science 96 (003) For a real function f dened on vertices of a graph G and S V, write f(s)= v S f(v) and f[v]=f(n [v]). A minus dominating function of G is dened in [3] as a function f V { 1; 0; 1} such that f[v] 1 for each v V. A signed dominating function of G is dened in [4] asf V { 1; 1} satisfying f[v] 1 for all v V. A minus (signed) dominating function f is minimal if every minus (signed) dominating function g satisfying g(v)6f(v) for every v V, is equal to f. It is easy to see that a minus dominating function is minimal if and only if for every vertex v V with f(v) 0, there exists a vertex u N[v] with f[u]=1 and a signed function is minimal if and only if every vertex v of weight 1, there exists some u N [v] such that f[u] =1 or. The minus domination number for a graph G is (G) = min{f(v ) f is a minimal minus dominating function}. Likewise, the signed domination number for a graph G is s (G) = min{f(v ) f is a minimal signed dominating function}. A majority dominating function of G is dened in [1] asf V { 1; 1} such that f[v] 1 for at least half the vertices of G, and the minimum weight of such a function is the majority domination number. For a positive integer k, a k-subdominating function (ksf) of G is a function f V { 1; 1} such that f[v]= u N (v) f(u) 1 for at least k vertices of G. The aggregate ag(f) of such a function is dened by ag(f)= v V f(v) and the k- subdomination number ks (G) by ks = min{ag(f) f is a ksf of G}. In the special cases k = V and k = V =, ks is respectively the signed domination number s (G) and the majority domination number maj (G). Since the problems of determining the signed domination number and minus domination number are NP-complete, many works on bounds for (G) and s (G) were studied in [,5 9,11]. In [3], the following conjecture was given. Conjecture 1 (Dunbar et al. [3]). If G is a bipartite graph of order n, then (G) 4( n +1 1) n.. Lower bounds on minus domination of a bipartite graph Theorem 1. If G =(X; Y ) is a bipartite graph of order n, then (G) 4( n +1 1) n. Proof. Let f be a minus dominating function of G satisfying f(v )= (G) and M = {v V f(v) = 1}; P = {v V f(v) =1}; Z = {v V f(v) =0} M X =M X; M Y =M Y; P X =P X; P Y =P Y; Z X =Z X; Z Y =Z Y; m x = M X ; m y = M Y ; p x = P X ; p y = P Y ; q x = Z X ; q y = Z Y

3 L. Kang et al. / Theoretical Computer Science 96 (003) Since f[v] 1 for every v V, we have N(v) P X for every v M Y.So e(p X ;M Y ) m y (1) For every v P X, N (v) M Y 6 N (v) P Y. Then e(p X ;M Y )= N (v) M Y v P X 6 v P X N (v) P Y 6 p x p y () By (1) and () we have m y 6 p x p y Similarly, m x 6 p x p y ; then m x + m y 6 p x p y (3) Since n = q x + q y + m x + m y + p x + p y and p x p y 6 p x + p y ; we have p x p y + m x + m y + q x + q y 6 n (4) Using (3) and (4) we have m x + m y + m x + m y + q x + q y 6 n (5) and m x + m y + m x + m y 6 n (6) By the denition, the inequalities can be deduced as follows (G) =f(v (G)) = p x + p y (m x + m y ) p x p y (m x + m y ) m x + m y (m x + m y ) (7)

4 9 L. Kang et al. / Theoretical Computer Science 96 (003) For notation convenience, we dene the following a = m x + m y ; h(y) =y +y (y 1); g(y) =y y (y 1) Since dh=dy =y +, dg=dy= y60, so h(y) is a monotonous increasing function and g(y) is a monotonous decreasing function. By (6) we have h(a)=a + a6n. And when y = 1+ 1+n, h(y)=( 1+ 1+n) +( 1+ 1+n) =1 1+n +1+n + 1+n = n So a n. By (7) we obtain (G) g(a) g( 1+ 1+n) =( 1+ 1+n) ( 1+ 1+n) =( 1+ 1+n) (1 1+n +1+n) =4( n +1 1) n We now show that this bound is best possible by the following graphs G construct by Dunbar et al. [3]. Let s 4 be an even integer, and let H be isomorphic to s= disjoint copies of K ;s. Let H 1 and H be two disjoint copies of H. Further, let X i and Y i be the sets of vertices of H i of degree and s, respectively, for i=1;. Now let G be the graph obtained from H 1 H by joining every vertex of Y 1 to every vertex of Y. Then G is a bipartite graph of order n=s(s + ) with partite sets X 1 Y and X Y 1. Let f be the function on G dened as follows let f(v)= 1ifv X 1 X, and let f(v)=1 if v Y 1 Y. Then it is easy to verify that f is a minus dominating function on G with (G)=f(V (G))=s s =4( n +1 1) n Theorem. If G =(X; Y ) is a bipartite graph of order n, then ( ) (G) n + ; 1 + max( X ; Y ) where X = min{d(v) v X }; Y = min{d(v) v Y }, and the bound is sharp.

5 L. Kang et al. / Theoretical Computer Science 96 (003) Proof. Let f; M X ;M Y ;P X ;P Y ;Z X ;Z Y ;m x ;m y ;q x ;q y ;p x and p y be dened as in the proof of Theorem 1. For any x V, let t x denotes the number of vertices of weight 0 in N(x). Then we have d(x) t x if x P d(x) 1 t N(x) M 6 x if x Z d(x) t x 1 if x M So d(y) = N (x) M Y + N(x) M Y + N(x) M Y y M Y x P X x Z X x M X 6 x P X d(x) t x = x X ( d(x) + x Z X d(x) 1 t x + ( d(x) tx x M X ) 1 t ) x 1 q x m x (8) Obviously, m y Y 6 y M Y d(y); (9) x X t x = 1 y Z Y d(y) 1 Y q y (10) Combining (8) (10) we obtain (q y +m y ) Y +(q x +m x ) 6 (11) Similarly, we have (q x +m x ) X +(q y +m y ) 6 (1) If q x +m x 6q y +m y,by(11) and (1) we have q x +m x 6 q y +m y 6 Y 1 + max( X ; Y ) ;

6 94 L. Kang et al. / Theoretical Computer Science 96 (003) So, (G) =n (q x +m x + q y +m y ) ( ) n + Y 1 + max( X ; Y ) ( ) n max( X ; Y ) If q y +m y q x +m x,by(11) and (1) we have So, q y +m y q x +m x 6 X 1 + max( X ; Y ) ; (G) =n (q x +m x + q y +m y ) ( ) n + X 1 + max( X ; Y ) ( ) n max( X ; Y ) In fact, this bound is sharp, it is easy to check that (K 1;k )=1= n =+=(1+max ( X ; Y )) 3. A lower bound on k-subdomination number of a graph The concept of k-subdomination was introduced by Cockayne and Mynhardt [1]. In [1], Cockayne and Mynhardt established a sharp lower bound on ks for trees. Moreover, they also gave a sharp lower bound on ks for trees if k6n= and proposed a conjecture. Theorem 3 (Cockayne and Mynhardt [1]). For any n-vertex tree T and integer k {1; ;;n}, ks 6(k +1) n. Conjecture (Cockayne and Mynhardt [1]). For any n-vertex tree and any k with 1 n k6n, ks6k n In [10], the conjecture was proved and a upper bound for a connected graph was given.

7 L. Kang et al. / Theoretical Computer Science 96 (003) Theorem 4 (Kang et al. [10]). For any connected graph of order n and any k with 1=n k6n, then k ks 6 (n k +1) n n k +1 In this section we give a lower bound for a graph G. Theorem 5. For any graph G of order n and size, ks n +(n k)( +) +1 Proof. Let f beak-subdominating function on G with f(v )= ks (G). Let P and M be the sets of vertices in G that are assigned the values 1 and 1, respectively. Then P + M =n and ks (G)= P M =n M. Furthermore, we let P 1 = {v P f[v] 1}; P = P P 1 ; M 1 = {v M f[v] 1}; M = M M 1 Clearly, P 1 + M 1 k. Since each vertex v of P 1 is adjacent to at most (1=)d(v) vertices of M, each vertex v of M 1 is adjacent to at most d(v)= 1 vertices of M. We have M 6 d(v) = M N (v) v M v V 6 d(v) + ( ) d(v) 1 + d(v) v P 1 v M 1 v P M = 1 d(v) M d(v) v V v P M 6 M + 1 v P M (d(v)+) 6 M +( P + M ) + As P + M 6n k, it follows that M 6 +(n k)( +) ( +1)

8 96 L. Kang et al. / Theoretical Computer Science 96 (003) Thus, ks = n M n +(n k)( +) +1 This completes the proof of Theorem 5. For the graphs in which each vertex has odd degree, the lower bound on ks Theorem 5 can be improved slightly. in Theorem 6. For every graph G in which each vertex has odd degree, ks n +(n k)( +) k +1 Proof. Let f; P; M; P 1 ;P ;M 1 and M be dened as in the proof of Theorem 5. Since every vertex of G has odd degree, it is easy to see that each vertex v of P 1 is adjacent to at most (d(v) 1)= vertices of M, each vertex v of M 1 is adjacent to at most (d(v) 1)= 1 vertices of M. Hence, we have M 6 d(v) = M N (v) v M v V 6 v P 1 d(v) 1 + v M 1 ( d(v) 1 ) 1 + d(v) v P M = 1 d(v) 1 ( P 1 + M 1 ) M d(v) v V v P M 6 1 ( P 1 + M 1 ) M + d(v)+ P M 6 1 ( P 1 + M 1 ) M +( P + M ) + Since P 1 + M 1 k; P + M 6n k, we have Hence, ( +1) M 6 k M 6 ( + )(n k) + +( + )(n k) k ( +1)

9 L. Kang et al. / Theoretical Computer Science 96 (003) Thus, ks (G) =n M +( + )(n k) k n +1 This completes the proof of Theorem 6. By Theorems 5 and 6, we easily obtain the following lower bounds on ks r-regular graphs. for Theorem 7. Let G be a r-regular graph of order n, then r + r +1 k n for r even; ks r +3 r +1 k n for r odd In the special cases where k = V and k = V =, Theorem 7 deduces to the following results. Corollary 7 (Henning [7]). For every r-regular graph G of order n, n for r odd; r +1 s (G) n for r even r +1 and the bounds are sharp. Corollary 8 (Henning [7]). For every r-regular (r ) graph G of order n, 1 r n for r odd; (r +1) maj (G) r n for r even (r +1) and the bounds are sharp. References [1] E.J. Cockayne, C.M. Mynhardt, On a generalization of signed dominating function of graphs, Ars Combin. 43 (1996) [] J.E. Dunbar, W. Goddard, S.T. Hedetniemi, M.A. Henning, A. McRae, The algorithm complexity of minus domination in graphs, Discrete Appl. Math. 68 (1996) [3] J.E. Dunbar, S.T. Hedetniemi, M.A. Henning, A. McRae, Minus domination in graphs, Discrete Math. 199 (1999)

10 98 L. Kang et al. / Theoretical Computer Science 96 (003) [4] J.E. Dunbar, S.T. Hedetniemi, M.A. Henning, P.J. Slater, Signed domination in graphs, in Y. Alavi, A. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, Wiley, New York, 1995, pp [5] T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Domination in Graphs Advanced Topics, Marcel Dekker, New York, [6] S.T. Hedetniemi, P.J. Slater, Fundamenters of Domination in Graphs, Marcel Dekker, New York, [7] M.A. Henning, Domination in regular graphs, Ars Combin. 43 (1996) [8] M.A. Henning, H.R. Hind, Strict majority functions on graphs, J. Graph Theorem 8 (1998) [9] M.A. Henning, P.J. Slater, Inequality relation domination parameters in cube graphs, Discrete Math. 158 (1996) [10] Kang Liying, Dang Chuangyin, Cai Maocheng, Shan Erfang, Upper bounds for k-subdomination number of graphs, Discrete Math. 47 (00) [11] Kang Liying, Shan Erfang, Lower bounds on dominating functions in graphs, Ars Combin. 56 (000)