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Some results on characterizing the edges of connected graphs with a given domination number
DISCRETE MATHEMATICS Discrete Mathematics 140 (1995) 149-166
ELSEVIER
Some results on characterizing the edges of connected graphs with a given domination number* Laura A. Sanchis Department of Computer Science, Colgate University, Hamilton, NY 13346, USA Received 1 July 1991; revised 2 October 1992
Abstract
A dominatin# set for a graph G = (V, E) is a subset of vertices V' c_ V such that for all v• V - V' there exists some u • V' for which {v,u} •E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let mz(G, D) denote the number of edges that have neither endpoint in D, and let m2(G,D ) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (mz(G,D),m2(G,D)) can attain for connected graphs having a given domination number.
1. Introduction
We denote by G =(V, E) an undirected graph with vertex set V= V(G) and edge set E=E(G). A dominating set for a graph G=(V, E) is a subset of vertices V'___V such that for all v e V - V ' , there exists some ue V' for which {v,u}~E. The domination number of G is the size of its smallest dominating set(s) and is denoted by ~,(G). The dominating set problem consists of finding the domination number for a given graph, and/or a dominating set of smallest cardinality. This problem has many practical applications. However, it is NP-hard [2]; it is therefore unlikely that an efficient algorithm for finding a smallest cardinality dominating set for an arbitrary graph can be found. One common approximation strategy for this problem consists of finding a dominating set of small (although not necessarily the smallest) cardinality, by selecting vertices with high degrees. It is therefore interesting to attempt to compare the degrees of dominating set vertices with those of vertices not in the dominating set, for dominating sets of smallest possible cardinality.
*Supported in part by National Science Foundation grant CCR-9101974. An abridged version of this paper appears in "Graph Theory, Combinatorics, and Algorithms:Proceedings of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs", John Wiley. 0012-365X/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved SSDI 0 0 1 2 - 3 6 5 X ( 9 3 ) E 0 1 8 0 - C
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Given a fixed minimum size dominating set D for a graph G=(V, E), the average degree for vertices in D, as well as the average degree for vertices in V-D, can be computed by determining how many edges join vertices in D, how many join vertices in V - D , and how many join a vertex in D to a vertex in V--D. The results in this paper provide one step in determining the possible values that these quantities can assume for connected graphs. Let G = (V, E) be any connected graph with n vertices, m edges, and dominating set D of size d. We denote by El (G, D) the set of edges of G that have neither endpoint in D, and by E 2(G, D) the set of edges of G that have at least one endpoint in D. Let ml (G, D)= IE~(G)I and m2(G, D)= Ig°(G)l . When it is clear from the context, we will use the abbreviations ml,m2. Clearly ml~<(n2 a) and (n-d)<~m2<~(a,)+d(n-d). In addition, in [5] it was shown that if ~,(G)=d~>3, then m=m~ +m2~<(*-~+1). In this paper we characterize the possible values that the pair (ml (G, D), m2 (G, D)) can attain. The paper is organized as follows. Section 2 examines one special type of graph, which is based on the extremal graphs that were shown to attain the upper bound in the number of edges in [5]. We specify the possible values that ml (G, D) and m2(G, D) can attain for this class of graphs. The main results, which are presented in Section 5, show that the restrictions on the values of m~(G, D) and m2 (G, D) which hold for the graphs discussed in Section 2, hold as well for all connected graphs G with ~(G)>~4, and for most connected graphs with ~(G)= 3. Sections 3 and 4 consist of preliminary results which are needed for the proofs in Section 5. Section 3 presents technical lemmas having to do with domination in graphs, while Section 4 contains a series of lemmas giving bounds on the values of ml(G,D) and m2(G,D) under various conditions. Section 6 provides related results for graphs with domination numbers 1 and 2.
2. One special type of graph The following lemma and theorem are from [4] and [5], respectively. [,emma 2.1 (Ore [4]). I f a graph G with n vertices has no isolated vertices, then (G) <.Nn/2.
Theorem 2.2. Let G be a graph with n vertices, domination number d where 3 <<.d <<.n/2, and no isolated vertices. Then the number of edges of G is at most (n-d2+1). I f G has exactly this number of edges then it must be of the following form: (1) An (n-d)-clique, together with an independent set of size d, such that each of the vertices in the (n-d)-clique is adjacent to exactly one of the vertices in the independent set, and such that each of these d vertices has at least one vertex adjacent to it. (2) For d = 3, G may consist of a clique of n - 5 vertices, together with 5 vertices Xl,Xz,X3,Xg, X5, with edges {xl,xa}, {x2,x4}, {x2,xs}, such that every vertex in the (n-5)-clique is adjacent to x4 and xs, and in addition adjacent to either xl or x 3. Moreover, at least one of these vertices is adjacent to xl and at least one to x3.
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For each pair (n, d) where 3 <~d <~n/2, denote by S., d the set of all pairs (G, D) where G is a graph of the type described in (1) above, and D is a dominating set for G of cardinality d. Given a graph G of this type, denote the vertices in the independent set by zl ..... Zd. For 1 ~2for 1 <~i<.d, 2k~
(n-~-k)<~m~<<.(n-~-k)+n-2d. (3) If k=d, then mx=("--22d)+(n--2d)=("-22d+l). However, this quantity equals ("-2d-k) with k = d - 1 , and is thus covered in the previous case. Clearly for all (G,D)eS,,d, m2(G, D)=("-a2+l)-ml(G, D). For 1 <~k<~d, define
ux(n,d, k ) = ( n - d 2 - k ),
u2(d,n, k ) = ( n - ~ - k ) + n - 2 d .
We have
ul(n,d,d-1)<~u2(n,d,d-1)
1)
~u2(n, d, 1 ) < ( n 2 d ). Note that ux(n, d, k - 1)-u2(n, d, k ) = d - k . So we have the following theorem.
Theorem 2.3. Let 3<<.d<<.n/2 and let (G,D)~S,,d. Then, one of the followino two conditions holds: (1) ml (G, D) =("2 d) and m2 (G, D ) = n - d . (2) There exists an inteoer k where 1 <.k <~d - 1 , such that ul(n, d, k) <.ml (G, D ) <~ u2(n, d, k) and m2(G,D)=(n-~ +1)--m I (G, D).
Moreover, if m'l and m'2 are integers satisfyin9 the conditions satisfied by ml (G, D) and m2 (G, D) in (1) or (2) above, then there exists a pair (G, D)eS,, d such that m'l = ml (G, D) and m'2 =mz(G, D). By judiciously removing edges from the graphs in S..a we can obtain connected graphs with the same domination number but fewer edges. Specifically, if (G, D)~S.. s and D=(wl .... , Wd}, define F ( D ) = [Jf=l {{w, wi}lwe V~--{w,}}. Denote by J-.,d the
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set of all pairs (G', D) where G' is a connected graph obtained by taking a pair
(G, D)eS.,d, and removing from G zero or more edges in E-F(D). The following theorem follows from the previous one and the definition of 3-., j. Theorem 2.4. Let 3 <<.d <<.n/2 and let (G,D)~3-.,4. Then (at least) one of the following
two conditions holds: (1) d - 1 <~ml(G, D)<~(~2a) and m2(G, D)=n--d. (2) There exists an integer k such that 1 <<.k ~ d - 1 and ml (G, D) <<.u2 (n, d, k), n-d<~m2(G,D)<~( n - d2+ l ~)-ul(n,d,k), n-I<~ml(G,D)+m2(G,D)<.(n-d+l) 2 " Moreover, if m'l and m'2 are integers satisfying the conditions satisfied by ml (G, D) and m2 (G, D) in (1) or (2) above, then there exists a pair (G, D)e3"., a such that m'l = ml (G, D) and m'2 = m2 (G, D). The following corollary makes more explicit the restrictions on the values of m 1 and m2 expressed in the above theorem. It is not hard to check that the restrictions given in the theorem and in the corollary are equivalent. Corollary 2.5. Let 3 <~d<~n/2 and let (G, D)~oq-.,d. Then all of the following conditions
hold: (1) n-d<~m2(G,D)<~(n-~2+l)-ul(n,d,d-1). (2) For 2<~k<~d- 1, if ml(G, D)>u2(n,d,k), then
[n-d+l) m2(G,D)<~ 2
-ul(n,d,k-1).
(3) If ml (G, D)>u2(n,d, 1), then m2(G, D ) = n - d . (4) n-- 1 <~ml(G, D)+m2(G, D)~<("-d2+l). Our goal is to show that the above restrictions apply not only to ~ , a, but to all pairs (G, D) where G is a connected graph with n vertices and domination number d >14, and D is a dominating set for G of size d. In addition, for d = 3, restrictions (1), (2), and (4) apply.
3. Technical lemmas Definition 3.1. Let G = (V, E) be a graph, and let (II1, ..., Vk) be a partition of V. For 1 ~
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(rl . . . . . rk)-dominates G if and only if there exists a dominating set S for G such that for each i, l <~i<<.k, IS~Vil=ri.
Definition 3.2. Let G = (V, E) be a graph and let (I,'1. . . . . Vk) be a partition of V. For 1 <~i<~k, let ri be an integer such that 1 ~
Observation 3.4. Let G = (V, E) be a graph, and let (V~ . . . . . Vk) be a partition of V. For l<<.i<~k, let ri be an integer such that l~I ID'I for each such D'. The result then follows by Hall's theorem on matchings of a bipartite graph, applied to the bipartite graph {D, IV} [3]. []
Lemma 3.6. Let G = (II, E) be a #raph with n >!3 vertices, m ed#es, and q >i 1 distinguished vertices x l . . . . . x~. Let r be such that 1 <<.r <~n - q - 1. Let W = V - { x l . . . . . x~ }. Suppose that G is ((W, r), ({x~ }, 1) . . . . . ({x~}, 1))-starred. Then m>>.rn+q-('+21). Moreover, if this bound is attained, then W contains a clique of r+ 1 vertices, every other vertex in W is adjacent to exactly r of the vertices in the clique and to no other vertices in G, and each xi is adjacent to all r + 1 vertices in the clique and to no other vertices in G. Proof. Let W1 consist of the vertices in W that are adjacent to each of the vertices xl . . . . . xq. Let W2 = W - W~. Let k = l WII. So I W21= n - q - k . Let G1 be the subgraph of G induced by WI. Since each group of at most r vertices in Wx must be adjacent to another vertex in W1, k>>.r+l, and hence GI is r-starred. This implies that ~(d~)~>r+ 1. By Vizing's theorem ([6], see also [1, Ch. 14]), the number of edges of
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G1 is at most L ( k - r (k2)
1 ) ( k - r + 1)/2J. Hence the number of edges of G1 is at least
(k-r-1)(k-r4.2 1)-2kr-k-r24"12
In addition, each vertex in g/x is adjacent to xl ..... x~ (kq edges), and each vertex in W2 must be adjacent to at least r vertices in W1 ( ( n - q - k ) r edges). Hence the number of edges of G is at least
( 2 k r - k - r 2 + l)/2 + k q + r ( n - q - k ) = r n + q -
( +r l2) + ( q - 1 / 2 ) ( k - r - 1 )
f r 4.1"\ >~rn+q--~ 2 )' since q >t 1 and k i> r + 1. If this bound is actually attained, then we must have k = r + 1, and hence the number of edges of GR is (,~1) and the graph is as described in the statement of the lemma. [] The following corollary follows by Observation 3.4. Corollary 3.7. Let G= (V,E) be a graph with n>~3 vertices, m edges, and q>>.1
distinguished
vertices xl ..... x~. Let r be such that l < ~ r ~ n - q - 1 . Let W= V-{x1 . . . . . x~}. If (IV, {Xx }. . . . . {x~}) does not (r, 1 . . . . . 1)-dominate G, then m ~ ( ~ ) - - /'n - - t -14 . zr+lx ~ 2 J. Moreover, if this upper bound is attained, then W contains an independent set oft 4- l vertices; all other vertices of W form a clique, and are each adjacent to one of the vertices in the independent set; and the vertices x 1.... , xqform a clique and are each adjacent to all vertices in I4I except for the r 4,1 independent set vertices. Lemma 3.8. Let G= (V, E) be a graph with n vertices and m edges, and let (Q1 . . . . . Qq)
be a partition of V, where 2 < . q ~ n - 1 . Let V = R I u R 2 , where R1 and R 2 are disjoint, and R 1 5 0 . Suppose that for each tuple (vl ..... vq), where vi~Qi and at least one vi belongs to R1, there exists w e V such that w is adjacent to vi for l <.i<.q. Then m>~n- 14. q - m a x ( l , 1R2[). Proof. The proof is by induction on n - q. The base case is n = q + 1. Assume without loss of generality that Q 1 = {x, y} and Qi = {zi} for i > 1. If for some i > 1, zi~R 1, then both x and y must be adjacent to all other vertices in V. This means that m>~n- 14. n - 2 > ~ n - 1 + q - 11>n- 1 + q - m a x ( I , [R2[). If on the other hand zieR2 for 2<~i<~q, then since R 1 5 0 , we may assume without loss of generality that x e R l . Then y must be adjacent to all other vertices, providing n - 1 edges. Ify is also in Rx, then there are n - 2 more edges and the argument is similar to the above. Otherwise, [R2[= q, and hence n - l = n - l + q - [ R 2 [ . For the induction step, assume n > q 4 . 1. If [R~[= 1, then n - 1 + q - [ R 2 [ =q, and since clearly m >t q, the result follows. So assume IRa[> 1. We consider the two cases IR2I---0 and ]R21 :/:0 separately.
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Case 1: IR2I=0. Assume without loss of generality that I Q I I > I . Let xeQ~. Let g equal the degree of x. If g = n - 1, choose Yie Qi for i > 1. Then x, Y2. . . . . yq must all be adjacent to some other vertex, implying the existence of q - 1 additional edges. Hence m >I n - 1 + q - 1. If g < n - 1, then let wx . . . . . wg be the vertices adjacent to x and let G' be the subgraph of G induced by V-{x}. Let Q'I=QI-{x}, Q;=Qi for i>1, R~= (Wx . . . . . wg}, and R~ = V - { x , wx ..... wg}. Note that R~ #0, and that for each tuple (vl .... , vq), where vieQi and at least one vi is in R], there exists we V - { x } such that w is adjacent to each v~. So by the induction hypothesis the number of edges of G' is at least n - 2 + q - 9 . Counting as well the edges incident on x, we have m>ln-l+q-1. Case 2: ]R2I>0. If any Qi consists entirely of RI vertices, then we may assume Case 1. So assume that each Q~ contains at least one R2 vertex. We call Q~ a singleton if it contains only one vertex (which must be in R2 by our assumption). We will consider the following two subcases: Case 2a: There exist vertices x, y such that x is not in a singleton Qi, yeR~, and x is not adjacent to y. Case 2b: No such vertices exist. Case 2a: Choose a pair of vertices x, y satisfying the stated conditions, choosing X 6 R 2 if possible. Note that if we must choose x e R x, then all vertices in R 2 belonging to Q~ that are not singletons must be adjacent to x, and that by our assumptions there exists at least one such R 2 vertex. Assume without loss of generality that xeQ~, so that IQa I> 1. Let S be the set of neighbors of x. Let G' be the subgraph of G induced by V - {x}. Let Q~ = Q1 - {x}, Q~ = Q, for i > 1, R~ = Sc~RI - {x}, and R~ = V - R~ - {x} = (R2wS)-{x}. Since yeRx, R ] ¢ 0 ; since q>~2, [R~I~>I. It is not hard to see that G', Q~ . . . . . Q'q, R~, R~ satisfy the assumptions of the lemma. By the induction hypothesis, the number of edges of G' is at least n - 2 + q - IR~ I. If xeR2, then [R'2]<~IR2[+[S[--1. If xeRa, then as noted above, Sc~R2¢0, and hence again IRWIn<[R21+ I S [ - 1 . Therefore counting as well the edges incident on x, we have m>~n--2+q--lR2l--lSl+l+lSl=n--l+q--lR2l. Case 2b: Let s be the number of singleton Q~, all belonging to R2 by our assumption. Since there exist no x, y satisfying the stated conditions, and since all vertices belonging to singleton Q~ must be adjacent to some vertex in a nonsingleton Q~, the number of edges of G is at least IR~ ](IR1 l - 1)/2+ IR~ l( IR2l-s)+s= IR~ I IR2I +(I R I I - 1)(I R~ I / 2 - s). Since IR2I~> q and n = IRII+IR2 l, this quantity is at least
n - l +q-lR21+ l - q - l R l l +lRxlq+(lRll-1)(lRll/2-s) = n - 1 + q - l R 2 l +(IRa l - 1 ) ( q - 1 + IRx I / 2 - s ) ~ n - 1 + q - l R 2 l , since we are assuming [RlI~>2, and s<~q.
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The first corollary below follows by letting R 2 = 0 . T h e second follows from O b s e r v a t i o n 3.4.
Corollary 3.9. Let G = (V, E) be a graph with n vertices and m edges, and let (QI . . . . . Q~) be a partition of V, where 2<.q<.n-1. If G is ((Qx,1),...,(Q~,l))-starred, then m>~n+q-2. Corollary 3.10. Let G= (V, E) be a graph with n vertices and m edges, and let (Q I ..... Qq) be a partition of V, where 2 <<.q <~n - 1. If(Q1 ..... Qq) does not (1 . . . . . 1)dominate G, then m < . ( ] ) - n - q + 2.
4. Preliminary results In this section we assume that G is a connected graph with n vertices and m edges, that ~(G) = d/> 3, and that D = {Vl . . . . . vd} is a m i n i m u m size d o m i n a t i n g set for G. Let W= V - D , and let H be the s u b g r a p h of G induced by W. N o t e that the set of edges of H is El (G, D). F o r each vertex w e W, let p(w) denote the n u m b e r of vertices in D that are adjacent to w. F o r 1 <<.k<<.d,define
s(n,d,k)=(n2d)-u2(n,d,k). It is not difficult to verify that
s(n,d,k)=(k- l)(n-cO+(d-k)-(k2) and that
(n-d2+ l )-rul ( n , d , k - 1 ) = k ( n - d ) - ( k 2 ) .
Lemma 4.1. Let l <<.k<<.d-1. If there exists a vertex w ~ W with p(w)>>.k+ l, then ml(G,D)~u2(n,d,k). Proof. Let p(w)=l~k+ 1. Assume without loss of generality that w is adjacent to vl . . . . . vz. By L e m m a 3.5 there exist distinct vertices Xz+x . . . . . xd in W adjacent to el + 1 . . . . . va, respectively. (The set {xl + 1. . . . . xd } is e m p t y if l = d.) Select (if l > k + 1) any l - k - 1 additional vertices Wk+2..... W~from W - - {w, xt+ 1. . . . . xd}. We consider first the case where k = 1. T h e ( d - 1) vertices w, w3 . . . . . wz, xl+ t . . . . . x~ cannot d o m i n a t e G; hence there must exist a vertex w' ~ W that is not adjacent to any of these vertices. So ml <<.(n2d)--(d--1)=u2(n,d, 1).
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For k~>2, let X = W--{w, wk+2..... W,, X,÷t ..... Xd}. Note that if (X, {w}, {Wk+2}, .... (Wi }, {Xz+ ~} . . . . . {Xd}) (k -- 1, 1, 1 . . . . . 1)-dominates H, then G has a dominating set of size d - 1, which is a contradiction. Hence by Corollary 3.7 k
mx~(n2d)-(k-1)(n-d)-(d-k)-4-(2)=(
n-
2d)-s(n,d,k)=u2(n,d,k).
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L e m m a 4.2. If p(w)=2 for all we W, then ml(G, D)<~u2(n,d,2). Proof. Let w be any vertex in W. Assume without loss of generality that w is adjacent to Vx and v2. By L e m m a 3.5 there exist distinct vertices x3 . . . . . xd such that x~ is adjacent to v~for 3 ~ i ~
[]
Lenuna 4.3. Let 1 <.k <.d-2. If at least one vertex in D is adjacent to k+ 1 other vertices in D, then ral (G, D) <<.u2 (n, d, k). Proof. Assume without loss of generality that vl is adjacent to v2 . . . . . vk+2. Suppose first that k ~
since n - d - k 2>~n-d-d>>.O. N o w let k = d - 2 . Since vl dominates D, ~(H)>~d-1. So by Vizing's theorem [6], mx <~L ( n - a - ( d -
1))(n-a-(d-
1)+2)/2 ]~<(n-2d+ 1)(n-2d+ 3)/2
=(n-2d+l)(n-2d+2)/2+(n-2d+l)/2=(n-2d+2)+(n-2d) - (n - 1 - 2d)/2 = u2 (n, d, d - 2 ) - (n - 1 - 2d)/2.
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I f d ~ < ( n - 1)/2, then mt <~u2(n,d,d-2). I f d > ( n - 1)/2, then n must be even and d=n/2, in which case
m~<~L( n - d - ( d - 1 ) ) ( n - d - ( d - 1 ) + 2 ) / 2
J=l=u2(n,d,d-2).
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Lemma 4.4. Let d >14. Suppose that there is a nonempty subset/3 ~_D such that each vertex in/3 is adjacent to at least two other vertices in/3. Then ml (G, D)<~u2(n, d, 2). Proof. Clearly 1/31>~3. Without loss of generality assume that vl, v2, v 3 ~/3 and that v2 is adjacent to vl and to va. Select distinct vertices x4 . . . . . Xae W, where xi is adjacent to v, for 4<~i<~d. Let X = W - { x 4 . . . . . xa}. If (X, {x,} . . . . . {xa}) (1, 1 . . . . . 1)-dominates H, then together with v2 we have a dominating set of size d - 1 for G, a contradiction. Hence by Corollary 3.7.
n-d If m 1 = (n ~ a) _ s (n, d, 2) + 1, then by the corollary there are two distinguished vertices in X (call them wl,w2), all other vertices in X are adjacent to each other and to x4 ..... xa, and each vertex in X - {wl, w2 } is adjacent to exactly one of w 1, w2. Choose y e X - { w l , w2}. Assume without loss of generality that y is adjacent to w2. If wl is adjacent to some vi where i > 3, or to v2, then {v2, y, v, ..... va} is a dominating set for G, a contradiction. If wt is adjacent to Vl, then since va must be adjacent to some other vis/3 besides v2, it follows that {vt, y, v4 . . . . . va} is a dominating set, again a contradiction. Similarly, if wl is adjacent to va, then {y, va . . . . . va} is a dominating set, a contradiction. Hence ml <~(n~a)-s(n,d, 2)=u2(n,d, 2). [] For the remaining lemmas, we will need the following additional notation. Let W' denote the set of vertices of W that are adjacent to only One vertex in D. Let b = [ W' [. Let D' consist of all vertices in D that are adjacent to some vertex in W'. Let b'= [D' [. Clearly b'<<,min(b,d). Without loss of generality assume that D'={vl . . . . . vv}. It will be convenient if b'~k(n-d)-(k)+ 1, then
WE W - - W '
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Hence
k(n-d-b)+kb-
~
p(w)<~
2 -l+m
+b
weW--g"
and
w~W_w(k-p(w))~(~) - 1 +m'-b(k-1). Define
f ( k , b ) = ( ~ ) - I +(b2)-b(k-1). If
p(w)<~k for
all we W, then
0<~ ~.~ (k-p(w))<~(~)-l+m'-b(k-1)<~f(k,b).
(1)
w~W--W'
Thus, under these assumptions, f(k,b) cannot be negative, and if f(k,b)=O, then for all w ~ W - W', b=b', and the vertices vl . . . . . Vb form a clique.
p(w)=k
Let 3 <<.k<,d. lf p(w)<~kfor all we W, and m2(G, D)>~k(n-d)-(k2)+ then b >/k + l.
Lemma 4.6.
1,
Proof. It can be easily checked that if b=k or b=k-1, then f(k,b)=-1. Hence b cannot equal either of these values. If b = k - 2, thenf(k, b) = 0, and hence all vertices in W - W ' are adjacent to k~>3 vertices in D. Let w~W-I4/'. This vertex must be adjacent to at least 2 vertices vi, vi in D - D'. So ( D - { v i , vj})u (w} is a dominating set for G, a contradiction. If b = k - 3 , then f ( k , b ) = 2. Hence, as can be seen from formula (1), all but at most 2 vertices in W - W' are adjacent to k vertices in D, and at most one can be adjacent to only k - 2 vertices in D. Because k ~ ~ 3. So there exists w e W such that w is adjacent to k = b + 3 vertices in D, and hence to at least 3 vertices in D - - ( e l . . . . . vb}. Assume without loss of generality that W is adjacent to Vb+ 1, Vb+ 2, V~+ 3" If in fact each vertex in W - W' is adjacent to k/> 3 vertices in D, then {el . . . . . Vb, Vb+ 1, W, Vb+4. . . . . V~} is a dominating set for G, a contradiction. If there is only one vertex y e W - W ' which is not adjacent to k vertices in D, then either y is adjacent to some vertex in {el .... ,Vb,Vb+4,...,Vd}, in which case again {el . . . . . Vb, Vb+1, W, Vb+4. . . . . Vd} is a dominating set, or else y is adjacent to at least one of Vb+l, Oh+2or vb+ a, in which case {el, ..., v~, w, Vb+j,Vb+4. . . . . Vd} where j = 1, 2 or 3, respectively, is a dominating set for G, again a contradiction. If neither of the above two cases hold, then there are two vertices Yl, Y2 e W - W', each adjacent to k - 1 >i 2 vertices in D. Again, it is possible to choose a vertex Vb+jwhere 1 ~
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Suppose 0 ~/4). Let t be equal to the n u m b e r of vertices in W - W' that are adjacent to 2 vertices in D. F o r each such vertex w, ( k - p ( w ) ) = k - 2 > _ . ( k - b - 2 ) . Hence by formula (1),
t(k-b-2)~f(k,b). Let k'= k - b . So 4 ~2, k' + 1 - 2t ~ k'/2, then it must be the case that t = (k' + 1)/2, and thus k ' + 1 - 2 t = 0 . So in all cases we have that t ' ( k ' - 1 - t ) < , ( k ' - 1 - t ) 2. Since k'~>4 and t <.(k' + l)/2, k ' - l - t >~l/2, so t' <.(k'- l - t ) , and t + t' <~k'- l = k - b - l ~ d - b - 1 and hence b + t + t ' ~ < d - 1 . Since n - d ~ d , this implies that there exists a vertex weW-W' adjacent to at least b + t + 2 vertices in D. So w must be adjacent to at least two vertices in D - { v l . . . . . vb+t}, say vb+t+l and vb+t+2, and {v~ . . . . . vb +t, w, Vb+t + a . . . . . Vd} is a d o m i n a t i n g set for G, which is a contradiction. So we must h a v e b > I k + l . [] L e m m a 4.7. Let 3 <<.k ~ d -
1. I f p(w) <~k for all w e W, m2(G, D)>>.k(n-d)-(~) + 1, and b = k + l, then ml(G,D)<.u2(n,d,k).
Proof. Becausef(k, k + 1)=0, b=b'. Hence each vertex in W' is adjacent vertex in {vl . . . . . vb}, the vertices Vl . . . . . vb form a clique, and each vertex adjacent to k i> 3 vertices in D. N o t e b/> 4. Suppose that there are two vertices x, y e W' that are adjacent. Assume of generality that x is adjacent to vl and y is adjacent to v~. Then {x, a d o m i n a t i n g set for G, a contradiction.
to a distinct in W - W' is without loss
v3 ..... Vd} is
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Suppose that some vertex w e W - W ' is adjacent to two vertices x, yEW', where again without loss of generality x is adjacent to vl and y is adjacent to v2. Then {w, va..... Vd} is a dominating set for G, a contradiction. Since neither of the above two conditions can hold, the vertices in W' must form an independent set, and each vertex in W - IV' can be adjacent to at most one vertex in IV'. Therefore
n-d
~<(2 sincen-d/>d.
)-s(n,d,k)=u2(n,d,k),
[]
Lemnm 4.8. Let k=d or k=d-1.
If p(w)<~k for all weW and m2(G,D)>>.
k(n-d)-(k)+ 1, then b<<.k+ 1. Proof. Recall that m' denotes the number of edges having both endpoints in D. We have
k(n-d)-(~)+l<~m2 <.m'+b+(n-d-b)k. So if b >i k + 2, then
m'>~b(k-1)-(~)+l ~(k+2)(k-1)-(k2)+1. If k = d, then
m'>.(d+2)(d-1)-d(d-1)/2+l = ( ~ ) + 2 d - 1 > ( ~ ) . If k = d - 1, then
m'>.(d+l)(d-2)-(d-1)(d-2)/2+l=(~)+d-2>(~). Since clearly m' ~ (g), we must have b ~
[]
Lenlraa 4.9. Let 3 ~k<~d-2. Ifp(w)<.kfor all we VV,m2(G, O)>~k(n-d)-(k)+ l, and b >~k+ 2, then rex(G, D)<.u2(n,d,k).
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162
Proof. As seen in the proof of Lemma 4.8,
m'>~b(k-1)-(k2)+l.
(2)
Suppose that each of the vertices in D' is adjacent to no more than k other vertices in D'. Then by Lemma 4.5, m ' ~~ b ' k/2 ~< .~ bk/2. Thus b ( k - 1 ) - ( ~ ) + 1 ~
b((k-2)/2)<~(k2)-l=(k-2)(k+l)/2. Therefore, since k t> 3, it follows that b ~/k + 2, there must be at least one vertex in D' that is adjacent to k + 1 other vertices in D. The result then follows from Lemma 4.3. []
Lemma 4.10. Let d>~4. I f p(w)<~2 for all w~W, m2(G,D)>>.2(n-d), and b>~3, then ml(G,D)<~u2(n,d,2). Proof. If b = 3, then f(2, b)=0, and therefore the vertices vl, v2, v3 form a clique. The result thus follows from Lemma 4.4. Assume b t>4. If there is some vertex in D' adjacent to 3 other vertices in D', the result follows by Lemma 4.3. Otherwise we have m'<~b', and since by formula (2) m'>~ b, it follows that b = b'. Moreover each of Vx. . . . . vb must be adjacent to two other vertices in D', and the result follows by Lemma 4.4. []
5. Main results We now give the main results showing that the conditions specified in Corollary 2.5 apply to all connected graphs with domination number at least 4.
Theorem 5.1. Let G be a connected graph with n vertices, ~ (G) = d >>.4, and minimum size dominating set D. l f m2(G, D)>>.n-d+ 1, then ml(G, D)<<.u2(n,d, 1). Proof. Let D = (Vl ..... Vd} and W = V--D. If there exists a vertex in W adjacent to at least 2 vertices in D, the result follows from Lemma 4.1. Suppose that each vertex in W is adjacent to only one vertex in D. Since m2 >t n - d + 1, there must be at least one edge joining two vertices in D. Assume without loss of generality that vl and v2 are adjacent. Select distinct vertices x3 . . . . . xa such that xi is adjacent to vi for 3<.i<~d. Since the vertices Vx,X3 .... ,Xd cannot dominate G, there exists a vertex we W - { x 3 . . . . . x~} not adjacent to any of these vertices. Similarly, since v2,xa ..... xa cannot dominate G, there exists a vertex
L.A. Sanchis/ Discrete Mathematics 140 (1995) 149-166 w ' e W - { x 3 . . . . . Xd} not adjacent to any of v2,x 3
.....
163
x d. If wv~w', then we have
ml <<.(n-2d)~2(d-2)<~(n2d)-(d-1). If on the other hand w = w', then w must be adjacent to some vjeD where j 1>3. Then since the set ({Vl, x3 . . . . . Xd}- {Xj/)U{Vj} cannot dominate G, there exists some other vertex in W not adjacent to any of these vertices. Hence
ml<~(n2d)-(d-2,-(d-3,<~(n2d)-(d-1).
[]
Theorem 5.2. Let G be a connected graph with n vertices, 7 (G) = d >>.4, and minimum size dominating set D. If m2(G, D)~>2(n--d), then ml (G, D)<<.u2(n,d, 2). Proof. If there is a vertex w e W such that p(w)>~3, then the result follows by L e m m a 4.1. Otherwise it must be the case that p(w)~<2 for all w e W. As in Section 4 let b be the number of vertices in W t h a t are adjacent to only one vertex in D. I f b = 0 , the result follows by L e m m a 4.2. Note that because m2>>.2(n-d), the number of edges joining vertices in D must be at least b. Hence by L e m m a 4.5, b cannot equal 1 or 2. If b~>3, the result follows by L e m m a 4.10. []
Theorem 5.3. Let G be a connected graph with n vertices, 7(G)=d>~4, and minimum size dominating set D. Let 3<~k<~d-1. If m2(G,D)>>.k(n-d)-(k2)+l, then m I (G, D)<~u2(n,d, k). Proof. If there is a vertex we W with p(w)>~k+l, then the result follows from L e m m a 4.1. Otherwise the result follows from Lemmas 4.6-4.9.
[]
Theorem 5.4. Let G be a connected graph with n vertices, y(G) = d >13,and minimum size dominating set D. Then m 2 (G, D) <<.(n-d2+1)_ Ul (n, d, d - 1). Proof. We assume that ml (G, D)>>.(n-~ + l ) - u l (n,d,d-1)+1 = d ( n - d ) - ( d 2 ) + 1, and show that this leads to a contradiction. Let k=d. Obviously we must have p(w)<~k=d for all we I41. Let b' and b be defined as in Section 4. By Lemmas 4.6 and 4.8, b = d + 1. However, if b = d + 1 = k + 1, then f(k, b) = 0, implying that b = b' <~d, which is impossible. []
Corollary 5.5. Let G be a connected graph with n vertices and 7(G)=d>~4. Let D be a dominating set for G of size d. Then all of the following conditions hold: (1) n-d<~m2(G,D)<~("-~+l)-ul(n,d,d-1).
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(2) For 2<~k<~d- 1, ifmt(G, D)>u2(n,d,k) then
m2(G,D)<~/ n - d 2+ 1 ) _ u l (n,d ' k - 1). (3) If ml(G,D)>u2(n,d, 1), then m2(G,D)=n-d. (4) n - 1 ~
Corollary 5.6. There exists a connected graph G with n vertices, ~(G)= 3 and minimum size dominating set D, such that ml(G,D)=m'l, and m2(G,D)=m'2, if and only if n - - i ~
6. Small domination numbers For the sake of completeness we give the analogous results for graphs with domination numbers 1 and 2. If~(G) = 1 and D is a dominating set of one vertex, it is obvious that m2(G, D ) = n - 1 and 0 ~
Theorem 6.1. Let G be a connected graph with n vertices, ~(G)=2, and minimum size dominating set D. Then
n - 1 ~mx (G, D)+m2(G, D)~
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165
Let D={vt,v2} and W = V - D . Let U denote the set of vertices in W that are adjacent to both vt and v2. Let p=IUI. No vertex in U can be adjacent to all other vertices in IV. Hence
If vl and v2 are not adjacent, then O<~p<~n-2, and m2=n-2+p. So
-F(m2-n+2)/27. If Vl and v2 are adjacent, then O<~p<~n-4, and m2=n-2+p+l. So (n z-22)
-F(m2-n+l)/27.
[]
7. Conclusions We have characterized the possible values for the pair (mx(G, D), m2(G,D)), for connected graphs G having minimum cardinality dominating sets D. The values that can be attained by these quantities are exactly those attained by the graphs in ~ . a for d=~,(G)~>4, and almost the same as those attained by graphs in ~ , 3 for ~(G)=3. Note that these results actually hold for all graphs with no isolated vertices. The only place where this hypothesis is explicitly invoked is in the proof of Lemma 3.5, which is then extensively used in the proofs in Section 4. As mentioned in the introduction, the set E2 (G, D) can be further subdivided into those edges having both endpoints in D, and those having only one endpoint in D. Let rh2(G, D) and th2 (G, D) denote the number of edges in these two groups, respectively. An additional question that can be investigated is what possible values the quantities ml (G, D), rha(G, D), and rh2(G, D) can attain in relation to each other. Characterizing these quantities would provide information about the relative average degrees for vertices in D and in V-D. It is not difficult to see that the graphs in ~ . a do not provide all the possible combinations of these values that can occur for arbitrary connected graphs. For example, consider the graph G1 = (V1, El) which is described as follows. Let G 1 have n vertices and let ),(G1)=d~>3. Let D1--{Vl ..... va} be a dominating set for G1. The vertices vt and v2 are adjacent and no other vertices in Dt are adjacent. Each vertex in II1 -D1 is adjacent to exactly one vertex in D1. For 1 ~
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also have th2(G1,D1)=l and 7n2(Gi,Dl)=n-d. But for all pairs th2 (G, D) >/1, then ml (G, D) ~
(G,D)e~.d, if
Acknowledgement The author is grateful to an anonymous referee for pointing out a better proof for Lemma 3.5.
References I-1] C. Berge, Graphs (North-Holland, Amsterdam 1985). [2-1 M. Garey and D.S. Johnson, Computers and Intractability - - A Guide to the Theory of NPCompleteness (Freeman, New York, 1979). [3] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26-30. [4] O. Ore, Theory of Graphs, American Mathematical Society Colloqium Publications, Vol. XXXVIII (American Mathematical Society, Providence, RI, 1962). [5] L.A. Sanchis, Maximum number of edges in connected graphs with a given domination number, Discrete Math. 87 (1991) 65-72. I-6] V.G. Vizing, A bound on the external stability number of a graph, Dokl. Akad. Nauk BSSR 164 (1965) 729-731.
ELSEVIER
Some results on characterizing the edges of connected graphs with a given domination number* Laura A. Sanchis Department of Computer Science, Colgate University, Hamilton, NY 13346, USA Received 1 July 1991; revised 2 October 1992
Abstract
A dominatin# set for a graph G = (V, E) is a subset of vertices V' c_ V such that for all v• V - V' there exists some u • V' for which {v,u} •E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let mz(G, D) denote the number of edges that have neither endpoint in D, and let m2(G,D ) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (mz(G,D),m2(G,D)) can attain for connected graphs having a given domination number.
1. Introduction
We denote by G =(V, E) an undirected graph with vertex set V= V(G) and edge set E=E(G). A dominating set for a graph G=(V, E) is a subset of vertices V'___V such that for all v e V - V ' , there exists some ue V' for which {v,u}~E. The domination number of G is the size of its smallest dominating set(s) and is denoted by ~,(G). The dominating set problem consists of finding the domination number for a given graph, and/or a dominating set of smallest cardinality. This problem has many practical applications. However, it is NP-hard [2]; it is therefore unlikely that an efficient algorithm for finding a smallest cardinality dominating set for an arbitrary graph can be found. One common approximation strategy for this problem consists of finding a dominating set of small (although not necessarily the smallest) cardinality, by selecting vertices with high degrees. It is therefore interesting to attempt to compare the degrees of dominating set vertices with those of vertices not in the dominating set, for dominating sets of smallest possible cardinality.
*Supported in part by National Science Foundation grant CCR-9101974. An abridged version of this paper appears in "Graph Theory, Combinatorics, and Algorithms:Proceedings of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs", John Wiley. 0012-365X/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved SSDI 0 0 1 2 - 3 6 5 X ( 9 3 ) E 0 1 8 0 - C
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Given a fixed minimum size dominating set D for a graph G=(V, E), the average degree for vertices in D, as well as the average degree for vertices in V-D, can be computed by determining how many edges join vertices in D, how many join vertices in V - D , and how many join a vertex in D to a vertex in V--D. The results in this paper provide one step in determining the possible values that these quantities can assume for connected graphs. Let G = (V, E) be any connected graph with n vertices, m edges, and dominating set D of size d. We denote by El (G, D) the set of edges of G that have neither endpoint in D, and by E 2(G, D) the set of edges of G that have at least one endpoint in D. Let ml (G, D)= IE~(G)I and m2(G, D)= Ig°(G)l . When it is clear from the context, we will use the abbreviations ml,m2. Clearly ml~<(n2 a) and (n-d)<~m2<~(a,)+d(n-d). In addition, in [5] it was shown that if ~,(G)=d~>3, then m=m~ +m2~<(*-~+1). In this paper we characterize the possible values that the pair (ml (G, D), m2 (G, D)) can attain. The paper is organized as follows. Section 2 examines one special type of graph, which is based on the extremal graphs that were shown to attain the upper bound in the number of edges in [5]. We specify the possible values that ml (G, D) and m2(G, D) can attain for this class of graphs. The main results, which are presented in Section 5, show that the restrictions on the values of m~(G, D) and m2 (G, D) which hold for the graphs discussed in Section 2, hold as well for all connected graphs G with ~(G)>~4, and for most connected graphs with ~(G)= 3. Sections 3 and 4 consist of preliminary results which are needed for the proofs in Section 5. Section 3 presents technical lemmas having to do with domination in graphs, while Section 4 contains a series of lemmas giving bounds on the values of ml(G,D) and m2(G,D) under various conditions. Section 6 provides related results for graphs with domination numbers 1 and 2.
2. One special type of graph The following lemma and theorem are from [4] and [5], respectively. [,emma 2.1 (Ore [4]). I f a graph G with n vertices has no isolated vertices, then (G) <.Nn/2.
Theorem 2.2. Let G be a graph with n vertices, domination number d where 3 <<.d <<.n/2, and no isolated vertices. Then the number of edges of G is at most (n-d2+1). I f G has exactly this number of edges then it must be of the following form: (1) An (n-d)-clique, together with an independent set of size d, such that each of the vertices in the (n-d)-clique is adjacent to exactly one of the vertices in the independent set, and such that each of these d vertices has at least one vertex adjacent to it. (2) For d = 3, G may consist of a clique of n - 5 vertices, together with 5 vertices Xl,Xz,X3,Xg, X5, with edges {xl,xa}, {x2,x4}, {x2,xs}, such that every vertex in the (n-5)-clique is adjacent to x4 and xs, and in addition adjacent to either xl or x 3. Moreover, at least one of these vertices is adjacent to xl and at least one to x3.
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For each pair (n, d) where 3 <~d <~n/2, denote by S., d the set of all pairs (G, D) where G is a graph of the type described in (1) above, and D is a dominating set for G of cardinality d. Given a graph G of this type, denote the vertices in the independent set by zl ..... Zd. For 1 ~2for 1 <~i<.d, 2k~
(n-~-k)<~m~<<.(n-~-k)+n-2d. (3) If k=d, then mx=("--22d)+(n--2d)=("-22d+l). However, this quantity equals ("-2d-k) with k = d - 1 , and is thus covered in the previous case. Clearly for all (G,D)eS,,d, m2(G, D)=("-a2+l)-ml(G, D). For 1 <~k<~d, define
ux(n,d, k ) = ( n - d 2 - k ),
u2(d,n, k ) = ( n - ~ - k ) + n - 2 d .
We have
ul(n,d,d-1)<~u2(n,d,d-1)
1)
~u2(n, d, 1 ) < ( n 2 d ). Note that ux(n, d, k - 1)-u2(n, d, k ) = d - k . So we have the following theorem.
Theorem 2.3. Let 3<<.d<<.n/2 and let (G,D)~S,,d. Then, one of the followino two conditions holds: (1) ml (G, D) =("2 d) and m2 (G, D ) = n - d . (2) There exists an inteoer k where 1 <.k <~d - 1 , such that ul(n, d, k) <.ml (G, D ) <~ u2(n, d, k) and m2(G,D)=(n-~ +1)--m I (G, D).
Moreover, if m'l and m'2 are integers satisfyin9 the conditions satisfied by ml (G, D) and m2 (G, D) in (1) or (2) above, then there exists a pair (G, D)eS,, d such that m'l = ml (G, D) and m'2 =mz(G, D). By judiciously removing edges from the graphs in S..a we can obtain connected graphs with the same domination number but fewer edges. Specifically, if (G, D)~S.. s and D=(wl .... , Wd}, define F ( D ) = [Jf=l {{w, wi}lwe V~--{w,}}. Denote by J-.,d the
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set of all pairs (G', D) where G' is a connected graph obtained by taking a pair
(G, D)eS.,d, and removing from G zero or more edges in E-F(D). The following theorem follows from the previous one and the definition of 3-., j. Theorem 2.4. Let 3 <<.d <<.n/2 and let (G,D)~3-.,4. Then (at least) one of the following
two conditions holds: (1) d - 1 <~ml(G, D)<~(~2a) and m2(G, D)=n--d. (2) There exists an integer k such that 1 <<.k ~ d - 1 and ml (G, D) <<.u2 (n, d, k), n-d<~m2(G,D)<~( n - d2+ l ~)-ul(n,d,k), n-I<~ml(G,D)+m2(G,D)<.(n-d+l) 2 " Moreover, if m'l and m'2 are integers satisfying the conditions satisfied by ml (G, D) and m2 (G, D) in (1) or (2) above, then there exists a pair (G, D)e3"., a such that m'l = ml (G, D) and m'2 = m2 (G, D). The following corollary makes more explicit the restrictions on the values of m 1 and m2 expressed in the above theorem. It is not hard to check that the restrictions given in the theorem and in the corollary are equivalent. Corollary 2.5. Let 3 <~d<~n/2 and let (G, D)~oq-.,d. Then all of the following conditions
hold: (1) n-d<~m2(G,D)<~(n-~2+l)-ul(n,d,d-1). (2) For 2<~k<~d- 1, if ml(G, D)>u2(n,d,k), then
[n-d+l) m2(G,D)<~ 2
-ul(n,d,k-1).
(3) If ml (G, D)>u2(n,d, 1), then m2(G, D ) = n - d . (4) n-- 1 <~ml(G, D)+m2(G, D)~<("-d2+l). Our goal is to show that the above restrictions apply not only to ~ , a, but to all pairs (G, D) where G is a connected graph with n vertices and domination number d >14, and D is a dominating set for G of size d. In addition, for d = 3, restrictions (1), (2), and (4) apply.
3. Technical lemmas Definition 3.1. Let G = (V, E) be a graph, and let (II1, ..., Vk) be a partition of V. For 1 ~
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153
(rl . . . . . rk)-dominates G if and only if there exists a dominating set S for G such that for each i, l <~i<<.k, IS~Vil=ri.
Definition 3.2. Let G = (V, E) be a graph and let (I,'1. . . . . Vk) be a partition of V. For 1 <~i<~k, let ri be an integer such that 1 ~
Observation 3.4. Let G = (V, E) be a graph, and let (V~ . . . . . Vk) be a partition of V. For l<<.i<~k, let ri be an integer such that l~
Lemma 3.6. Let G = (II, E) be a #raph with n >!3 vertices, m ed#es, and q >i 1 distinguished vertices x l . . . . . x~. Let r be such that 1 <<.r <~n - q - 1. Let W = V - { x l . . . . . x~ }. Suppose that G is ((W, r), ({x~ }, 1) . . . . . ({x~}, 1))-starred. Then m>>.rn+q-('+21). Moreover, if this bound is attained, then W contains a clique of r+ 1 vertices, every other vertex in W is adjacent to exactly r of the vertices in the clique and to no other vertices in G, and each xi is adjacent to all r + 1 vertices in the clique and to no other vertices in G. Proof. Let W1 consist of the vertices in W that are adjacent to each of the vertices xl . . . . . xq. Let W2 = W - W~. Let k = l WII. So I W21= n - q - k . Let G1 be the subgraph of G induced by WI. Since each group of at most r vertices in Wx must be adjacent to another vertex in W1, k>>.r+l, and hence GI is r-starred. This implies that ~(d~)~>r+ 1. By Vizing's theorem ([6], see also [1, Ch. 14]), the number of edges of
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G1 is at most L ( k - r (k2)
1 ) ( k - r + 1)/2J. Hence the number of edges of G1 is at least
(k-r-1)(k-r4.2 1)-2kr-k-r24"12
In addition, each vertex in g/x is adjacent to xl ..... x~ (kq edges), and each vertex in W2 must be adjacent to at least r vertices in W1 ( ( n - q - k ) r edges). Hence the number of edges of G is at least
( 2 k r - k - r 2 + l)/2 + k q + r ( n - q - k ) = r n + q -
( +r l2) + ( q - 1 / 2 ) ( k - r - 1 )
f r 4.1"\ >~rn+q--~ 2 )' since q >t 1 and k i> r + 1. If this bound is actually attained, then we must have k = r + 1, and hence the number of edges of GR is (,~1) and the graph is as described in the statement of the lemma. [] The following corollary follows by Observation 3.4. Corollary 3.7. Let G= (V,E) be a graph with n>~3 vertices, m edges, and q>>.1
distinguished
vertices xl ..... x~. Let r be such that l < ~ r ~ n - q - 1 . Let W= V-{x1 . . . . . x~}. If (IV, {Xx }. . . . . {x~}) does not (r, 1 . . . . . 1)-dominate G, then m ~ ( ~ ) - - /'n - - t -14 . zr+lx ~ 2 J. Moreover, if this upper bound is attained, then W contains an independent set oft 4- l vertices; all other vertices of W form a clique, and are each adjacent to one of the vertices in the independent set; and the vertices x 1.... , xqform a clique and are each adjacent to all vertices in I4I except for the r 4,1 independent set vertices. Lemma 3.8. Let G= (V, E) be a graph with n vertices and m edges, and let (Q1 . . . . . Qq)
be a partition of V, where 2 < . q ~ n - 1 . Let V = R I u R 2 , where R1 and R 2 are disjoint, and R 1 5 0 . Suppose that for each tuple (vl ..... vq), where vi~Qi and at least one vi belongs to R1, there exists w e V such that w is adjacent to vi for l <.i<.q. Then m>~n- 14. q - m a x ( l , 1R2[). Proof. The proof is by induction on n - q. The base case is n = q + 1. Assume without loss of generality that Q 1 = {x, y} and Qi = {zi} for i > 1. If for some i > 1, zi~R 1, then both x and y must be adjacent to all other vertices in V. This means that m>~n- 14. n - 2 > ~ n - 1 + q - 11>n- 1 + q - m a x ( I , [R2[). If on the other hand zieR2 for 2<~i<~q, then since R 1 5 0 , we may assume without loss of generality that x e R l . Then y must be adjacent to all other vertices, providing n - 1 edges. Ify is also in Rx, then there are n - 2 more edges and the argument is similar to the above. Otherwise, [R2[= q, and hence n - l = n - l + q - [ R 2 [ . For the induction step, assume n > q 4 . 1. If [R~[= 1, then n - 1 + q - [ R 2 [ =q, and since clearly m >t q, the result follows. So assume IRa[> 1. We consider the two cases IR2I---0 and ]R21 :/:0 separately.
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Case 1: IR2I=0. Assume without loss of generality that I Q I I > I . Let xeQ~. Let g equal the degree of x. If g = n - 1, choose Yie Qi for i > 1. Then x, Y2. . . . . yq must all be adjacent to some other vertex, implying the existence of q - 1 additional edges. Hence m >I n - 1 + q - 1. If g < n - 1, then let wx . . . . . wg be the vertices adjacent to x and let G' be the subgraph of G induced by V-{x}. Let Q'I=QI-{x}, Q;=Qi for i>1, R~= (Wx . . . . . wg}, and R~ = V - { x , wx ..... wg}. Note that R~ #0, and that for each tuple (vl .... , vq), where vieQi and at least one vi is in R], there exists we V - { x } such that w is adjacent to each v~. So by the induction hypothesis the number of edges of G' is at least n - 2 + q - 9 . Counting as well the edges incident on x, we have m>ln-l+q-1. Case 2: ]R2I>0. If any Qi consists entirely of RI vertices, then we may assume Case 1. So assume that each Q~ contains at least one R2 vertex. We call Q~ a singleton if it contains only one vertex (which must be in R2 by our assumption). We will consider the following two subcases: Case 2a: There exist vertices x, y such that x is not in a singleton Qi, yeR~, and x is not adjacent to y. Case 2b: No such vertices exist. Case 2a: Choose a pair of vertices x, y satisfying the stated conditions, choosing X 6 R 2 if possible. Note that if we must choose x e R x, then all vertices in R 2 belonging to Q~ that are not singletons must be adjacent to x, and that by our assumptions there exists at least one such R 2 vertex. Assume without loss of generality that xeQ~, so that IQa I> 1. Let S be the set of neighbors of x. Let G' be the subgraph of G induced by V - {x}. Let Q~ = Q1 - {x}, Q~ = Q, for i > 1, R~ = Sc~RI - {x}, and R~ = V - R~ - {x} = (R2wS)-{x}. Since yeRx, R ] ¢ 0 ; since q>~2, [R~I~>I. It is not hard to see that G', Q~ . . . . . Q'q, R~, R~ satisfy the assumptions of the lemma. By the induction hypothesis, the number of edges of G' is at least n - 2 + q - IR~ I. If xeR2, then [R'2]<~IR2[+[S[--1. If xeRa, then as noted above, Sc~R2¢0, and hence again IRWIn<[R21+ I S [ - 1 . Therefore counting as well the edges incident on x, we have m>~n--2+q--lR2l--lSl+l+lSl=n--l+q--lR2l. Case 2b: Let s be the number of singleton Q~, all belonging to R2 by our assumption. Since there exist no x, y satisfying the stated conditions, and since all vertices belonging to singleton Q~ must be adjacent to some vertex in a nonsingleton Q~, the number of edges of G is at least IR~ ](IR1 l - 1)/2+ IR~ l( IR2l-s)+s= IR~ I IR2I +(I R I I - 1)(I R~ I / 2 - s). Since IR2I~> q and n = IRII+IR2 l, this quantity is at least
n - l +q-lR21+ l - q - l R l l +lRxlq+(lRll-1)(lRll/2-s) = n - 1 + q - l R 2 l +(IRa l - 1 ) ( q - 1 + IRx I / 2 - s ) ~ n - 1 + q - l R 2 l , since we are assuming [RlI~>2, and s<~q.
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The first corollary below follows by letting R 2 = 0 . T h e second follows from O b s e r v a t i o n 3.4.
Corollary 3.9. Let G = (V, E) be a graph with n vertices and m edges, and let (QI . . . . . Q~) be a partition of V, where 2<.q<.n-1. If G is ((Qx,1),...,(Q~,l))-starred, then m>~n+q-2. Corollary 3.10. Let G= (V, E) be a graph with n vertices and m edges, and let (Q I ..... Qq) be a partition of V, where 2 <<.q <~n - 1. If(Q1 ..... Qq) does not (1 . . . . . 1)dominate G, then m < . ( ] ) - n - q + 2.
4. Preliminary results In this section we assume that G is a connected graph with n vertices and m edges, that ~(G) = d/> 3, and that D = {Vl . . . . . vd} is a m i n i m u m size d o m i n a t i n g set for G. Let W= V - D , and let H be the s u b g r a p h of G induced by W. N o t e that the set of edges of H is El (G, D). F o r each vertex w e W, let p(w) denote the n u m b e r of vertices in D that are adjacent to w. F o r 1 <<.k<<.d,define
s(n,d,k)=(n2d)-u2(n,d,k). It is not difficult to verify that
s(n,d,k)=(k- l)(n-cO+(d-k)-(k2) and that
(n-d2+ l )-rul ( n , d , k - 1 ) = k ( n - d ) - ( k 2 ) .
Lemma 4.1. Let l <<.k<<.d-1. If there exists a vertex w ~ W with p(w)>>.k+ l, then ml(G,D)~u2(n,d,k). Proof. Let p(w)=l~k+ 1. Assume without loss of generality that w is adjacent to vl . . . . . vz. By L e m m a 3.5 there exist distinct vertices Xz+x . . . . . xd in W adjacent to el + 1 . . . . . va, respectively. (The set {xl + 1. . . . . xd } is e m p t y if l = d.) Select (if l > k + 1) any l - k - 1 additional vertices Wk+2..... W~from W - - {w, xt+ 1. . . . . xd}. We consider first the case where k = 1. T h e ( d - 1) vertices w, w3 . . . . . wz, xl+ t . . . . . x~ cannot d o m i n a t e G; hence there must exist a vertex w' ~ W that is not adjacent to any of these vertices. So ml <<.(n2d)--(d--1)=u2(n,d, 1).
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For k~>2, let X = W--{w, wk+2..... W,, X,÷t ..... Xd}. Note that if (X, {w}, {Wk+2}, .... (Wi }, {Xz+ ~} . . . . . {Xd}) (k -- 1, 1, 1 . . . . . 1)-dominates H, then G has a dominating set of size d - 1, which is a contradiction. Hence by Corollary 3.7 k
mx~(n2d)-(k-1)(n-d)-(d-k)-4-(2)=(
n-
2d)-s(n,d,k)=u2(n,d,k).
[]
L e m m a 4.2. If p(w)=2 for all we W, then ml(G, D)<~u2(n,d,2). Proof. Let w be any vertex in W. Assume without loss of generality that w is adjacent to Vx and v2. By L e m m a 3.5 there exist distinct vertices x3 . . . . . xd such that x~ is adjacent to v~for 3 ~ i ~
[]
Lenuna 4.3. Let 1 <.k <.d-2. If at least one vertex in D is adjacent to k+ 1 other vertices in D, then ral (G, D) <<.u2 (n, d, k). Proof. Assume without loss of generality that vl is adjacent to v2 . . . . . vk+2. Suppose first that k ~
since n - d - k 2>~n-d-d>>.O. N o w let k = d - 2 . Since vl dominates D, ~(H)>~d-1. So by Vizing's theorem [6], mx <~L ( n - a - ( d -
1))(n-a-(d-
1)+2)/2 ]~<(n-2d+ 1)(n-2d+ 3)/2
=(n-2d+l)(n-2d+2)/2+(n-2d+l)/2=(n-2d+2)+(n-2d) - (n - 1 - 2d)/2 = u2 (n, d, d - 2 ) - (n - 1 - 2d)/2.
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I f d ~ < ( n - 1)/2, then mt <~u2(n,d,d-2). I f d > ( n - 1)/2, then n must be even and d=n/2, in which case
m~<~L( n - d - ( d - 1 ) ) ( n - d - ( d - 1 ) + 2 ) / 2
J=l=u2(n,d,d-2).
[]
Lemma 4.4. Let d >14. Suppose that there is a nonempty subset/3 ~_D such that each vertex in/3 is adjacent to at least two other vertices in/3. Then ml (G, D)<~u2(n, d, 2). Proof. Clearly 1/31>~3. Without loss of generality assume that vl, v2, v 3 ~/3 and that v2 is adjacent to vl and to va. Select distinct vertices x4 . . . . . Xae W, where xi is adjacent to v, for 4<~i<~d. Let X = W - { x 4 . . . . . xa}. If (X, {x,} . . . . . {xa}) (1, 1 . . . . . 1)-dominates H, then together with v2 we have a dominating set of size d - 1 for G, a contradiction. Hence by Corollary 3.7.
n-d If m 1 = (n ~ a) _ s (n, d, 2) + 1, then by the corollary there are two distinguished vertices in X (call them wl,w2), all other vertices in X are adjacent to each other and to x4 ..... xa, and each vertex in X - {wl, w2 } is adjacent to exactly one of w 1, w2. Choose y e X - { w l , w2}. Assume without loss of generality that y is adjacent to w2. If wl is adjacent to some vi where i > 3, or to v2, then {v2, y, v, ..... va} is a dominating set for G, a contradiction. If wt is adjacent to Vl, then since va must be adjacent to some other vis/3 besides v2, it follows that {vt, y, v4 . . . . . va} is a dominating set, again a contradiction. Similarly, if wl is adjacent to va, then {y, va . . . . . va} is a dominating set, a contradiction. Hence ml <~(n~a)-s(n,d, 2)=u2(n,d, 2). [] For the remaining lemmas, we will need the following additional notation. Let W' denote the set of vertices of W that are adjacent to only One vertex in D. Let b = [ W' [. Let D' consist of all vertices in D that are adjacent to some vertex in W'. Let b'= [D' [. Clearly b'<<,min(b,d). Without loss of generality assume that D'={vl . . . . . vv}. It will be convenient if b'
WE W - - W '
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Hence
k(n-d-b)+kb-
~
p(w)<~
2 -l+m
+b
weW--g"
and
w~W_w(k-p(w))~(~) - 1 +m'-b(k-1). Define
f ( k , b ) = ( ~ ) - I +(b2)-b(k-1). If
p(w)<~k for
all we W, then
0<~ ~.~ (k-p(w))<~(~)-l+m'-b(k-1)<~f(k,b).
(1)
w~W--W'
Thus, under these assumptions, f(k,b) cannot be negative, and if f(k,b)=O, then for all w ~ W - W', b=b', and the vertices vl . . . . . Vb form a clique.
p(w)=k
Let 3 <<.k<,d. lf p(w)<~kfor all we W, and m2(G, D)>~k(n-d)-(k2)+ then b >/k + l.
Lemma 4.6.
1,
Proof. It can be easily checked that if b=k or b=k-1, then f(k,b)=-1. Hence b cannot equal either of these values. If b = k - 2, thenf(k, b) = 0, and hence all vertices in W - W ' are adjacent to k~>3 vertices in D. Let w~W-I4/'. This vertex must be adjacent to at least 2 vertices vi, vi in D - D'. So ( D - { v i , vj})u (w} is a dominating set for G, a contradiction. If b = k - 3 , then f ( k , b ) = 2. Hence, as can be seen from formula (1), all but at most 2 vertices in W - W' are adjacent to k vertices in D, and at most one can be adjacent to only k - 2 vertices in D. Because k ~
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Suppose 0 ~/4). Let t be equal to the n u m b e r of vertices in W - W' that are adjacent to 2 vertices in D. F o r each such vertex w, ( k - p ( w ) ) = k - 2 > _ . ( k - b - 2 ) . Hence by formula (1),
t(k-b-2)~f(k,b). Let k'= k - b . So 4 ~
1. I f p(w) <~k for all w e W, m2(G, D)>>.k(n-d)-(~) + 1, and b = k + l, then ml(G,D)<.u2(n,d,k).
Proof. Becausef(k, k + 1)=0, b=b'. Hence each vertex in W' is adjacent vertex in {vl . . . . . vb}, the vertices Vl . . . . . vb form a clique, and each vertex adjacent to k i> 3 vertices in D. N o t e b/> 4. Suppose that there are two vertices x, y e W' that are adjacent. Assume of generality that x is adjacent to vl and y is adjacent to v~. Then {x, a d o m i n a t i n g set for G, a contradiction.
to a distinct in W - W' is without loss
v3 ..... Vd} is
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Suppose that some vertex w e W - W ' is adjacent to two vertices x, yEW', where again without loss of generality x is adjacent to vl and y is adjacent to v2. Then {w, va..... Vd} is a dominating set for G, a contradiction. Since neither of the above two conditions can hold, the vertices in W' must form an independent set, and each vertex in W - IV' can be adjacent to at most one vertex in IV'. Therefore
n-d
~<(2 sincen-d/>d.
)-s(n,d,k)=u2(n,d,k),
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Lemnm 4.8. Let k=d or k=d-1.
If p(w)<~k for all weW and m2(G,D)>>.
k(n-d)-(k)+ 1, then b<<.k+ 1. Proof. Recall that m' denotes the number of edges having both endpoints in D. We have
k(n-d)-(~)+l<~m2 <.m'+b+(n-d-b)k. So if b >i k + 2, then
m'>~b(k-1)-(~)+l ~(k+2)(k-1)-(k2)+1. If k = d, then
m'>.(d+2)(d-1)-d(d-1)/2+l = ( ~ ) + 2 d - 1 > ( ~ ) . If k = d - 1, then
m'>.(d+l)(d-2)-(d-1)(d-2)/2+l=(~)+d-2>(~). Since clearly m' ~ (g), we must have b ~
[]
Lenlraa 4.9. Let 3 ~k<~d-2. Ifp(w)<.kfor all we VV,m2(G, O)>~k(n-d)-(k)+ l, and b >~k+ 2, then rex(G, D)<.u2(n,d,k).
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Proof. As seen in the proof of Lemma 4.8,
m'>~b(k-1)-(k2)+l.
(2)
Suppose that each of the vertices in D' is adjacent to no more than k other vertices in D'. Then by Lemma 4.5, m ' ~~ b ' k/2 ~< .~ bk/2. Thus b ( k - 1 ) - ( ~ ) + 1 ~
b((k-2)/2)<~(k2)-l=(k-2)(k+l)/2. Therefore, since k t> 3, it follows that b ~
Lemma 4.10. Let d>~4. I f p(w)<~2 for all w~W, m2(G,D)>>.2(n-d), and b>~3, then ml(G,D)<~u2(n,d,2). Proof. If b = 3, then f(2, b)=0, and therefore the vertices vl, v2, v3 form a clique. The result thus follows from Lemma 4.4. Assume b t>4. If there is some vertex in D' adjacent to 3 other vertices in D', the result follows by Lemma 4.3. Otherwise we have m'<~b', and since by formula (2) m'>~ b, it follows that b = b'. Moreover each of Vx. . . . . vb must be adjacent to two other vertices in D', and the result follows by Lemma 4.4. []
5. Main results We now give the main results showing that the conditions specified in Corollary 2.5 apply to all connected graphs with domination number at least 4.
Theorem 5.1. Let G be a connected graph with n vertices, ~ (G) = d >>.4, and minimum size dominating set D. l f m2(G, D)>>.n-d+ 1, then ml(G, D)<<.u2(n,d, 1). Proof. Let D = (Vl ..... Vd} and W = V--D. If there exists a vertex in W adjacent to at least 2 vertices in D, the result follows from Lemma 4.1. Suppose that each vertex in W is adjacent to only one vertex in D. Since m2 >t n - d + 1, there must be at least one edge joining two vertices in D. Assume without loss of generality that vl and v2 are adjacent. Select distinct vertices x3 . . . . . xa such that xi is adjacent to vi for 3<.i<~d. Since the vertices Vx,X3 .... ,Xd cannot dominate G, there exists a vertex we W - { x 3 . . . . . x~} not adjacent to any of these vertices. Similarly, since v2,xa ..... xa cannot dominate G, there exists a vertex
L.A. Sanchis/ Discrete Mathematics 140 (1995) 149-166 w ' e W - { x 3 . . . . . Xd} not adjacent to any of v2,x 3
.....
163
x d. If wv~w', then we have
ml <<.(n-2d)~2(d-2)<~(n2d)-(d-1). If on the other hand w = w', then w must be adjacent to some vjeD where j 1>3. Then since the set ({Vl, x3 . . . . . Xd}- {Xj/)U{Vj} cannot dominate G, there exists some other vertex in W not adjacent to any of these vertices. Hence
ml<~(n2d)-(d-2,-(d-3,<~(n2d)-(d-1).
[]
Theorem 5.2. Let G be a connected graph with n vertices, 7 (G) = d >>.4, and minimum size dominating set D. If m2(G, D)~>2(n--d), then ml (G, D)<<.u2(n,d, 2). Proof. If there is a vertex w e W such that p(w)>~3, then the result follows by L e m m a 4.1. Otherwise it must be the case that p(w)~<2 for all w e W. As in Section 4 let b be the number of vertices in W t h a t are adjacent to only one vertex in D. I f b = 0 , the result follows by L e m m a 4.2. Note that because m2>>.2(n-d), the number of edges joining vertices in D must be at least b. Hence by L e m m a 4.5, b cannot equal 1 or 2. If b~>3, the result follows by L e m m a 4.10. []
Theorem 5.3. Let G be a connected graph with n vertices, 7(G)=d>~4, and minimum size dominating set D. Let 3<~k<~d-1. If m2(G,D)>>.k(n-d)-(k2)+l, then m I (G, D)<~u2(n,d, k). Proof. If there is a vertex we W with p(w)>~k+l, then the result follows from L e m m a 4.1. Otherwise the result follows from Lemmas 4.6-4.9.
[]
Theorem 5.4. Let G be a connected graph with n vertices, y(G) = d >13,and minimum size dominating set D. Then m 2 (G, D) <<.(n-d2+1)_ Ul (n, d, d - 1). Proof. We assume that ml (G, D)>>.(n-~ + l ) - u l (n,d,d-1)+1 = d ( n - d ) - ( d 2 ) + 1, and show that this leads to a contradiction. Let k=d. Obviously we must have p(w)<~k=d for all we I41. Let b' and b be defined as in Section 4. By Lemmas 4.6 and 4.8, b = d + 1. However, if b = d + 1 = k + 1, then f(k, b) = 0, implying that b = b' <~d, which is impossible. []
Corollary 5.5. Let G be a connected graph with n vertices and 7(G)=d>~4. Let D be a dominating set for G of size d. Then all of the following conditions hold: (1) n-d<~m2(G,D)<~("-~+l)-ul(n,d,d-1).
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(2) For 2<~k<~d- 1, ifmt(G, D)>u2(n,d,k) then
m2(G,D)<~/ n - d 2+ 1 ) _ u l (n,d ' k - 1). (3) If ml(G,D)>u2(n,d, 1), then m2(G,D)=n-d. (4) n - 1 ~
Corollary 5.6. There exists a connected graph G with n vertices, ~(G)= 3 and minimum size dominating set D, such that ml(G,D)=m'l, and m2(G,D)=m'2, if and only if n - - i ~
6. Small domination numbers For the sake of completeness we give the analogous results for graphs with domination numbers 1 and 2. If~(G) = 1 and D is a dominating set of one vertex, it is obvious that m2(G, D ) = n - 1 and 0 ~
Theorem 6.1. Let G be a connected graph with n vertices, ~(G)=2, and minimum size dominating set D. Then
n - 1 ~mx (G, D)+m2(G, D)~
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Let D={vt,v2} and W = V - D . Let U denote the set of vertices in W that are adjacent to both vt and v2. Let p=IUI. No vertex in U can be adjacent to all other vertices in IV. Hence
If vl and v2 are not adjacent, then O<~p<~n-2, and m2=n-2+p. So
-F(m2-n+2)/27. If Vl and v2 are adjacent, then O<~p<~n-4, and m2=n-2+p+l. So (n z-22)
-F(m2-n+l)/27.
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7. Conclusions We have characterized the possible values for the pair (mx(G, D), m2(G,D)), for connected graphs G having minimum cardinality dominating sets D. The values that can be attained by these quantities are exactly those attained by the graphs in ~ . a for d=~,(G)~>4, and almost the same as those attained by graphs in ~ , 3 for ~(G)=3. Note that these results actually hold for all graphs with no isolated vertices. The only place where this hypothesis is explicitly invoked is in the proof of Lemma 3.5, which is then extensively used in the proofs in Section 4. As mentioned in the introduction, the set E2 (G, D) can be further subdivided into those edges having both endpoints in D, and those having only one endpoint in D. Let rh2(G, D) and th2 (G, D) denote the number of edges in these two groups, respectively. An additional question that can be investigated is what possible values the quantities ml (G, D), rha(G, D), and rh2(G, D) can attain in relation to each other. Characterizing these quantities would provide information about the relative average degrees for vertices in D and in V-D. It is not difficult to see that the graphs in ~ . a do not provide all the possible combinations of these values that can occur for arbitrary connected graphs. For example, consider the graph G1 = (V1, El) which is described as follows. Let G 1 have n vertices and let ),(G1)=d~>3. Let D1--{Vl ..... va} be a dominating set for G1. The vertices vt and v2 are adjacent and no other vertices in Dt are adjacent. Each vertex in II1 -D1 is adjacent to exactly one vertex in D1. For 1 ~
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also have th2(G1,D1)=l and 7n2(Gi,Dl)=n-d. But for all pairs th2 (G, D) >/1, then ml (G, D) ~
(G,D)e~.d, if
Acknowledgement The author is grateful to an anonymous referee for pointing out a better proof for Lemma 3.5.
References I-1] C. Berge, Graphs (North-Holland, Amsterdam 1985). [2-1 M. Garey and D.S. Johnson, Computers and Intractability - - A Guide to the Theory of NPCompleteness (Freeman, New York, 1979). [3] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26-30. [4] O. Ore, Theory of Graphs, American Mathematical Society Colloqium Publications, Vol. XXXVIII (American Mathematical Society, Providence, RI, 1962). [5] L.A. Sanchis, Maximum number of edges in connected graphs with a given domination number, Discrete Math. 87 (1991) 65-72. I-6] V.G. Vizing, A bound on the external stability number of a graph, Dokl. Akad. Nauk BSSR 164 (1965) 729-731.