2-connected graphs with small 2-connected dominating sets

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1 2-connecte graphs with small 2-connecte ominating sets Yair Caro, Raphael Yuster 1 Department of Mathematics, University of Haifa at Oranim, Tivon 36006, Israel Abstract Let G be a 2-connecte graph. A subset D of V (G) is a 2-connecte ominating set if every vertex of G has a neighbor in D an D inuces a 2-connecte subgraph. Let γ 2 (G) enote the minimum size of a 2-connecte ominating set of G. Let (G) be the minimum egree of G. For an n-vertex graph G, we prove that ln (G) γ 2 (G) n (G) (1 + o (1)) where o (1) enotes a function that tens to 0 as. The upper boun is asymptotically tight. This extens the results in [3,5,10,12]. Key wors: Domination, Connectivity, Minimum egree 1 Introuction All graphs consiere here are finite, unirecte, an simple. For stanar graph-theoretic terminology the reaer is referre to [4]. A subset D of vertices in a graph G is a ominating set if every vertex not in D has a neighbor in D. The omination number, enote γ(g), is the minimum size of a ominating set. A graph G with more than r vertices is r-connecte if eleting any set of at most r 1 vertices results in a connecte graph. If G is an r-connecte graph, then G has a ominating set that inuces an r-connecte subgraph (simply take the whole graph as a ominating set). Such a ominating set is calle an r-connecte ominating set. Let γ r (G) enote the minimum size of an r-connecte ominating set of G. The parameter γ 1 (G) is also calle the connecte omination number of G. Note that for r 1, every r-connecte 1 Corresponing Author. aress: raphy@research.haifa.ac.il Preprint submitte to Elsevier 28 February 2012

2 ominating set is a strong ominating set, namely, every vertex of G (whether in the ominating set or not) is ominate. The problem of fining small ominating sets is an active topic of research in the area of graph algorithms an combinatorics (see e.g. [8,9]). In particular, upper bouns as a function of the minimum egree of the graph are well stuie. A classic result prove inepenently by Lovász [10] (see another proof in [2]), Arnautov [3], an Payan [12] states that γ(g) n 1+ln(+1) +1 for every n-vertex graph G with minimum egree. This result is asymptotically optimal for general graphs. Alon [1] prove by probabilistic methos that when n is large there exists a -regular graph with no ominating set of size less than (1 + o(1)) 1+ln(+1) +1 n. (For 3, exact results were obtaine in [11,13]). For connecte omination, Caro, West, an Yuster [5] showe by more complicate arguments that the boun obtaine by Lovász, Payan an Arnautov also hols in a much more restricte case, namely γ 1 (G) n ln (1 + o (1)). They also supplie a sequential eterministic algorithm that prouces a connecte ominating set with (at most) this size, in polynomial time. Thus, it is interesting to etermine whether the boun of Lovász, Payan, an Arnautov also hols for r-connecte ominating sets, for every fixe r. Namely: Conjecture 1 Let r be a fixe positive integer. If G is an r-connecte graph with n vertices an minimum egree then γ r (G) n ln (1 + o (1)). The result of Caro, West an Yuster shows that the conjecture hols for r = 1. In this paper we prove it for r = 2. Theorem 2 If G is a 2-connecte graph with n vertices an minimum egree, then γ 2 (G) n ln (1 + o (1)). Notice that both Conjecture 1 an Theorem 2 are relevant (an interesting) only for sufficiently large. The proof of the 2-connecte case turns out to be more complicate than the proof for the case r = 1. This is partly ue to the fact that γ 1 (G)/γ(G) is boune by the constant 3 (this is shown in [6] an also in [5]). However, one cannot boun γ 2 (G)/γ(G) by any constant (see example in Section 2). There are also other obstacles when consiering the 2-connecte case. The upper boun in Theorem 2 is asymptotically sharp. This is ue to the fact that γ r (G) γ(g) an that the construction of Alon mentione above for γ(g) yiels, for every fixe r, an r-connecte graph (assuming that n an are large enough). In Section 2 we introuce the require tools neee for the proof of Theorem 2. The proof itself, which uses the probabilistic metho, appears in Section 3. 2

3 2 Preliminary Lemmas We start with three lemmas that are require for the proof of Theorem 2. In orer to present the first lemma, we nee to state several efinitions. A k- ominating set of a graph G is a subset X V (G) such that every v outsie X has at least k neighbors in X. A block of a graph H is a maximal subgraph B of H such that B has no cut-vertex. Every ege of H belongs to precisely one block. A block with three or more vertices is a 2-connecte subgraph. It is an elementary exercise to see that if H is not a block then at least one block of H (in fact, at least two) has precisely one cut-vertex of H. Such a block is calle a leaf block. For a subset X of vertices of G, let G[X] enote the subgraph of G inuce by X. We begin with a lemma that shows that any 2-ominating set (with at least three vertices) of a 2-connecte graph can be extene into a 2-connecte ominating set by an iterative process that reuces the number of blocks until there is only one. Lemma 3 Let G be a 2-connecte graph. If X is a 2-ominating set of G (with X > 2), an G[X] has s blocks, then there exists a 2-connecte ominating set of G, containing X, whose size is less than X + 10(s 1). Proof: Let t enote the number of components of G[X]. Clearly, 1 t s. We shall consier the weight function w efine by w(x) = 4t+s. We will show that, as long as s > 1, we can fin at most two vertices outsie X such that aing them to X ecreases w. Hence, by aing at most 8t+2s 10 10(s 1) vertices, we get w = 5 an s = t = 1. In particular, there exists a 2-connecte ominating set of G. Consier first the case t > 1. Thus, we have X = A B such that no ege of G connects a vertex in A with a vertex in B. Let a A an b B be two vertices whose istance in G is the smallest possible. This istance is at most 3, since otherwise there is a vertex in the mile of the shortest path between a an b that is not ominate by X. Consier first the case where a an b have a common neighbor, c, outsie X. Aing c to X ecreases the number of components by some p 1, but it may increase the number of blocks (by aing cut-eges) by at most p + 1. Hence, w(x {c}) 4(t p) + (s + p + 1) 4t + s 2. Now consier the case where the istance in G between a an b is 3. Let (a, c,, b) be a shortest path from a to b; note that c, / X. Aing c an to X ecreases the number of components by some p 1, but may increase the number of blocks by at most p + 2. Hence, w(x {c, }) 4(t p) + (s + p + 2) 4t + s 1. Now consier the case t = 1. If s = 1 we are one. Otherwise, let U be the vertex set of a leaf block in G[X], an let u be the unique cut-vertex of G[X] 3

4 belonging to U. Let a U \{u} an b X\U be two vertices whose istance in G u is the smallest possible. This istance is finite since G u is connecte. We claim also that this istance is at most 3. Inee, otherwise, there is a vertex in the mile of the shortest path between a an b in G u having no neighbor in X \ {u}, contraicting the fact that X is a 2-ominating set. Aing the one or two vertices on this shortest path between a an b to X ecreases the number of blocks an therefore ecreases w. Remark: Note that the requirement that X is a 2-ominating set is neee. There are examples showing that γ 2 (G)/γ(G) may be arbitrary large. Here is one: Take a cycle on the vertices v 1,..., v n, where n 7. Connect v 3 to all the other vertices except for v 1 an v 5. The resulting graph is 2-connecte (it is Hamiltonian). The omination number is 3, since v 1, v 3, v 5 is a ominating set, an two vertices cannot ominate everything. However, every 2-connecte ominating set has at least n 4 vertices. The secon tool we nee is a special case of a theorem of Kouier an Lonc [7]. Lemma 4 (Kouier, Lonc [7]) The vertex set of an n-vertex graph with minimum egree can be covere with at most n/ subgraphs such that each is a vertex, an ege, or a cycle. The thir lemma shows that if a graph has many vertices of (relatively) high egree, then it also has a large subgraph whose minimum egree is relatively high. Lemma 5 Fix ɛ an with 1/2 > ɛ > 0 an 1, an let G be a graph with n vertices. If at most ɛn vertices have egree less than, then G has a set of at least (1 3ɛ)n vertices inucing a subgraph with minimum egree greater than /4. Proof: Since ɛ < 0.5, we have that G has more than n/4 eges. It is well known that every graph with n vertices an αn eges has a subgraph with minimum egree at least α (see, e.g. [4] p. xvii). Hence, G has a subgraph with minimum egree greater than /4. Let Q be the largest set of vertices of a subgraph of G with minimum egree greater than /4. Let q = Q an let X be the vertices outsie Q. By the maximality of Q, every vertex of X has at most /4 neighbors in Q, an every subgraph of G[X] has a vertex of egree at most /4. Hence, G[X] has at most (n q)/4 eges. Furthermore, X contains at least n q ɛn vertices whose egree in G is at least. Thus, the sum of the egrees (in G[X]) is at least (n q ɛn)3/4. Hence, we must have (n q)/4 (n q ɛn)3/8. Thus, q (1 3ɛ)n. 4

5 3 Proof of the main result The basic iea of the proof of Theorem 2 is the following. We choose a ranom subset X of vertices of the graph. Given X, we (eterministically) a to it the subset Y of all vertices that have only a few (or no) neighbors in X. Clearly, X Y is a ominating set. If necessary, we a to X Y a few more vertices in orer to make it a 2-connecte ominating set. The crucial argument is that with high probability, this process results in a 2-connecte ominating set that is not too large. The etaile proof follows. Proof of Theorem 2: Fix ɛ (0,.5). We shall prove that, for sufficiently large, every 2-connecte n-vertex graph G with minimum egree has a 2-connecte ominating set of size at most ( ɛ)n ln. Let p = (1 + ɛ) ln, an let X be a ranom set of vertices in G, where each vertex is chosen inepenently with probability p. Let Y be the set of vertices in G that have fewer than k neighbors in X, where k = ln. Note that X Y is a k-ominating set of G. Let H = G[X Y ], an let s be the number of blocks of H. Accoring to Lemma 3, we can a at most 10(s 1) vertices to X Y an obtain a 2-connecte ominating set of G. With X = x an Y = y, it follows that γ 2 (G) < x + y + 10s. (1) To obtain the esire upper boun on γ 2 (G), we first prove an upper boun on s in terms of other parameters. In the egree sequence of G[X], liste in nonecreasing orer, let 1 be the term at position ɛx. Thus G[X] has at most ɛx vertices with egree less than. By Lemma 5, X has a subset Q of size at least (1 3ɛ)x such that (G[Q]) > /4. By Lemma 4, Q can be covere using at most 4 Q / subgraphs of G[Q] that are cycles, eges, or vertices. Aing the x Q + y vertices of (X Q) Y as 1-vertex subgraphs yiels a covering of V (H) using r such subgraphs of H, where r 4x/ + 3ɛx + y. We claim that s, the number of blocks of H, is at most 2r 1. Enlarge the subgraphs in the covering to become blocks of H. This may combine some subgraphs, but in any case we obtain a covering of V (H) using at most r blocks of H. This implies that H has at most 2r 1 blocks, using the fact that if the vertices of a graph G are covere by at most r blocks in G, then G has at most 2r 1 blocks. (Since blocks are connecte, the union of the initial covering blocks has at most r components. The vertices of an omitte block lie in istinct components, so aing the block reuces the number of components, an this can happen at most r 1 times.) We have shown that s 8 x + 6ɛx + 2y. 5

6 It now follows from (1) that γ 2 (G) < (1 + 60ɛ)x + 21y + 80 x. (2) We shall boun the expectations of the summans in (2). Obviously, E [x] = pn = (1 + ɛ) n ln. (3) Examining any neighbors of a vertex v in G yiels an upper boun on the probability that v has fewer than k neighbors in X. Thus, Pr[x Y ] k 1 i=0 ( ) k 1 p i (1 p) i < i i=0 (p) i e p( k) = (4) which is at most o( 1 ), so O ( k(2 ln ) k (1+ɛ/2)), ( ) n E[y] = o. (5) The estimation of E[ x ] is somewhat more elicate. Using conitional expectation, we split the computation into three parts. Let A be the event that x > 3np. Let B be the event that x 3np an < k. Let C be the event that x 3np an k. Hence, Thus, [ ] x E = E [ ] x E [ x A ] Pr[A] + E [ ] [ ] x x B Pr[B] + E C Pr[C] n Pr[x > 3np] + 3np Pr[ < k] + 3np k. (6) We nee to boun the two probabilities Pr[x > 3np] an Pr[ < k]. Since x has binomial istribution B(n, p), we can use large eviation inequalities to boun Pr[x > 3np]. We use the inequality of Chernoff (cf. [2] Appenix A) which states that for every β 1, Pr[x > βpn] < ( e β 1 β β) pn. Putting β = 3 an using the inequality ln 27 2(1 + ɛ) > 1 yiels ( ) e 2 pn ( ) e 2 (1+ɛ) ln Pr[x > 3np] < < < (7) 6

7 By (4) the probability that a vertex v of G has fewer than k neighbors in X is at most o( 1 ). The event that v is chosen for X is inepenent of the number of neighbors it has in X (since there are no loops in the graph). Thus, for sufficiently large, Pr[v X an G[x] (v) < k] < p 1. Let z enote the number of vertices in G[X] with egree less than k. By the last inequality E[z] < np/. By Markov s inequality, Pr[z > 0.5ɛnp] = Pr[z > 0.5ɛ(np/)] < 2 ɛ. Let q = 1 2/(ɛ) Pr[x < 0.6np]. Hence, z 0.5ɛnp an x 0.6np with probability at least q. In this situation, z ɛx 1. Thus, with probability at least q, the element at position ɛx in the egree sequence of G[X] has value at least k. It follows that Pr[ k + 1] q an therefore Pr[ k] 1 q = 2 + Pr[x < 0.6np]. (8) ɛ As before, we can boun Pr[x < 0.6np] using large eviation inequalities. We shall use the inequality of Chernoff (cf. [2]) appenix A) which states that for every a > 0, Pr[x np < a] < exp In particular, for a = 0.4np we get ( a2 2pn Pr[x < 0.6np] < exp( 0.08pn) < exp( 0.08 ln ) = Plugging the last inequality into (8) we get Pr[ k] < 2 ɛ + 1. (9) 0.08 ). By (6), (7), an (9), [ ] x E n 1 ( 2 + 3np ɛ + 1 ) + 3np 0.08 k. (10) Notice that (10) yiels E [ ] x = o(n ln /). By (2), (3), (5), it hols for sufficiently large that [ ] x γ 2 (G) = E[γ 2 (G)] < (1 + 60ɛ)E[x] + 21E[y] + 80E 7

8 (1 + 60ɛ) (1 + ɛ) n ln ( ) ( n + 21o + 80o n ln ) < ( ɛ)n ln. Remark: The proof of theorem 2 in fact gives a 2-connecte k-ominating set of size at most n ln (1 + o (1)) whenever k ln. References [1] N. Alon, Transversal numbers of uniform hypergraphs, Graphs Comb. 6 (1990), 1 4. [2] N. Alon an J. H. Spencer, The Probabilistic Metho, John Wiley an Sons Inc., New York, [3] V.I. Arnautov, Estimation of the exterior stability number of a graph by means of the minimal egree of the vertices (in Russian), Prikl. Mat. i Programmirovanie 11 (1974), 3 8. [4] B. Bollobás, Extremal Graph Theory, Acaemic Press, Lonon, [5] Y. Caro, D. West an R. Yuster, Connecte omination an spanning trees with many leaves, Siam J. Disc. Math. 13 (2000), [6] P. Duchet an H. Meyniel, On Hawiger s number an stability numbers, Annal. Disc. Math. 13 (1982), [7] M. Kouier an Z. Lonc, Covering cycles an k-term egrees sums, Combinatorica 16 (1996), [8] T. Haynes, S.T. Heetniemi an P. Slater, Domination in Graphs: The Theory, Marcel Dekker Publishers, New York, [9] T. Haynes, S.T. Heetniemi an P. Slater, Domination in Graphs: Selecte Topics, Marcel Dekker Publishers, New York, [10] L. Lovász, On the ratio of optimal an integral fractional covers, Disc. Math. 13 (1975), [11] W. McCuaig an B. Shepher, Domination in graphs with minimum egree two, J. Graph Theory 13 (1989), [12] C. Payan, Sur le nombre absorption un graphe simple (French), Proc. Colloq. Th. Graphes (Paris 1974), C.C.E.R.O. 17 (1975), [13] B. Ree, Paths, stars an the number three, Combinatorics, Probability an Computing 5 ( 1996),