Generallowerboundonthe sizeof (H; k)-stable graphs

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1 J Comb Optim (015) 9: DOI /s y Generallowerboundonthe sizeof (H; )-stable graphs Andrzej Ża Published online: 15 February 013 The Author(s) 013. This article is published with open access at Springerlin.com Abstract A graph G is called (H; )-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its vertices. By stab(h; ) we denote the minimum size among the sizes of all (H; )-vertex stable graphs. In this paper we present a first (non-trivial) general lower bound for stab(h; ) with regard to the order, connectivity and minimum degree of H. This bound is nearly sharp for = 1. Keywords Vertex-stable graph Minimum degree Connectivity 1 Introduction By a word graph we mean a simple graph in which multiple edges (but not loops) are allowed. Given a graph G, V (G) denotes the vertex set of G and E(G) denotes the edge set of G. Furthermore, G := V (G) is the order of G and G := E(G) is the size of G. ByN G (x) we denote the set of vertices adjacent with x in G. Fora vertex set X, thesetn G (X) denotes the external neighbourhood of X in G, i.e. N G (X) ={y V (G) \ X : y is adjacent with some x X}. Consider the following problem. Suppose that we want to build a construction having certain properties using elements from sets S 1,..., S t. Each element from a set S i has a given cost c i. Thus, the total cost (depending only on the numbers and costs of used elements) of every construction can be computed. We will consider this ind of problem in case where the feasible constructions are graphs with certain properties and accesible elements are vertices and edges. A. Ża (B) AGH University of Science and Technology, Al. Miciewicza 30, Kraów, Poland zaandrz@agh.edu.pl 13

2 368 J Comb Optim (015) 9: We require that a feasible (from our point of view) graph G (feasible construction) contains a given subgraph H. In fact, we require more. Some sensors may get damaged, hence, we want that even if some of them are spoiled, the special configuration of sensors and connections is still assured in the net. Clearly, we want to assure this configuration with minimal cost. The following problem has attracted some attention recently. Let H be any graph and a non-negative integer. A graph G is called (H; ) ver texstable (in short (H; ) stable)if G contains a subgraph isomorphic to H ever after removing any of its vertices. Then stab(h; ) denotes minimum size among the sizes of all (H; )-vertex stable graphs. Note that if H does not have isolated vertices then after adding to or removing from a (H; )-vertex stable graph any number of isolated vertices we still have a (H; )-vertex stable graph with the same size. Therefore, in the sequel we assume that no graph in question has isolated vertices. The notion of (H; )-vertex stable graphs was introduced in Dude et al. (008)(an edge version of this notion was also considered, see Franl and Katona 008; Horváth and Katona 011). So far the exact value of stab(h; ) for any is nown in the case when H = C 3, C 4, K 4, K 1,m (Dude et al. 008), H = K 5 (Fouquet et al. 01a), and H = K q with sufficiently large (Ża 011). On the other hand, for small the value stab(h; ) is nown when H = K m,n and = 1, see Dude and Zwone (009) and Dude and Ża (010), and when H = K n and n/ + 1, see Fouquet et al. (01b). In all the above cases minimum vertex stable graphs are characterized. Furthermore, stab (C n ; 1) is nown for infinitely many n s and for remaining n s it has one of only two possible values, see Cichacz et al. (011). An upper and a lower bound on stab (C n ; ) for sufficiently large n is also presented therein. Our aim in this paper is to prove a more general result. Namely, we give a lower bound for the size of a (H; ) stable graph, where H is an arbitrary graph. The bound depends on the order, connectivity and minimum degree of H. This generalizes a similar lower bound obtained for = 1 only in Cichacz et al. (01). Theorem 1 Let H be a graph of order n, minimal degree δ 1 and connectivity κ 1. If δ (n + 1 κ) δκn + (1+δ δκ) then stab(h; ) δ n + δκn (δκ δ 1) (1) In particular, Theorem 1 leads to a new lower bound for stab (C n ; ) which, for, is signifficantly better than the one obtained in Cichacz et al. (011). Namely, Corollary If n then stab(c n ; ) n + n. In Sect. 3 we present a family of graphs for which our new lower bound gives reasonable estimates. On the other hand, for regular graphs our bound gives 13 stab(h; ) H + δκ H (δκ + δ + 1),

3 J Comb Optim (015) 9: which in many cases is signifficantly better than the trivial bound stab(h; ) H + (H). Proof of Theorem 1 We start with the following inequality which will be used later Proposition 3 If, l, m are positive integers with m + l, then m(m 1)...(m + 1) (m l)(m l 1)...(m l + 1) m + (l 1). () m Proof The proof is by induction on.for = 1 the assertion is easy to chec. Assume that > 1 and the inequality is true for 1. Then m(m 1)...(m +)(m +1) (m l)(m l 1)...(m l +)(m l +1) m+( 1)(l 1) m +1 m +1 m l +1 = m+( 1)(l 1) m l +1 = 1 + Recall the following observation. by the induction hypothesis l m l m l = m+(l 1) m. Proposition 4 (Dude et al. (008)) Let δ H be a minimal degree of a graph H. Then in any (H; )-vertex stable graph G with minimum size, deg G v δ H for each vertex v G. Proof of Theorem 1 Let G be a (H; ) stable graph with minimum size and let G =v. Let S ={x 1,..., x m } V (G) be a set of vertices of degree greater than or equal to δ + 1inG. By Proposition 4 all other vertices of G have degree δ. LetC 1,..., C q be components of G S.Let G be a graph that arises from G by contracting every edge of each C i, i = 1,..., q. Suppose first that G contains a vertex u S with at most κ 1 neighbors in G. Consider a component C in G that corresponds to u. Since G is minimal (H; )- stable, every vertex of C is contained in some copy of H. So, consider a copy of H that contains at least one vertex of C. Note, that this copy of H may contain only vertices from C and the vertices which are neighbors of u in G, because, otherwise H contains a cutting set of cardinality less than κ. Thus, C contains at least n + 1 κ vertices, C n + 1 κ. (3) Note that after removing from G any vertex x i N G (C) each vertex of C is not any longer a vertex of H. Indeed, after removing x i, its neighbors in C G have degree less than δ. Thus, they cannot be in H. Hence, their neighbors in C G would have degrees less than δ in H. Thus, the latter vertices connot be in H neither, and so on. Therefore, 13

4 370 J Comb Optim (015) 9: since G is (H; ) stable, G C contains a copy of H. Thus, G C H nδ. Hence, by (3) and by the assumption on n, G nδ + C δ nδ + δ (n + 1 κ) δ n + δκn (δκ δ 1). (4) Therefore, assume that every vertex u V ( G) \ S has at least κ neighbors in G. We may assume that m + κ. Indeed, otherwise by removing vertices from S,we remove at least one vertex x N G (C) for each component C {C 1,..., C q }. Hence, all remaining vertices (possibly except κ 1 < n vertices from S) become useless for H. We will randomly delete exactly vertices from S. Hence, each -tuple is removed with probability ( m) 1. In the resulting graph some vertices may have degree less than δ and so do not belong to any copy of H.Foru V (G) \ S let X u denote the indicator random variable with value 1 if u belongs to some copy of H. Since w V ( G) \ S has at least κ neighbors in G, the probability that u V (G) \ S can be used in some copy of H is less than or equal to (m κ ) ( m ). Hence, E(X u ) ) ( m ). ( m κ Thus, the expected value of vertices from V (G) \ S that can be used in a copy of H is Hence, E E u V (G)\S u V (G)\S X u X u = u V (G)\S E(X u ) (v m)( m κ) ( m ). (m κ)...(m κ + 1) (v m) m...(m + 1) m (v m) m + (κ 1), (5) by inequality (). Thus, there are vertices in S such that after deleting them we can m use in H at most m + (v m) vertices of G. Therefore, m+(κ 1) 13 m m + (v m) n, and so m + (κ 1) m + (κ 1) v m (n m + ) m

5 J Comb Optim (015) 9: Hence, G δ(v m)+(δ+1)m δ m+(κ 1) (n m + ) m + δ+1 m =: f (m) It is not difficult to chec (by examining the derivative) that the function f (x) = δ x+(κ 1) (n x + ) x + δ+1 x, x + l, has minimum in x 0 = κδn +. Hence, G f (x 0 ) = δ n + δκn (δκ δ 1) 3 Question of tightness For κ, δ, n, N with κ δ n + 1let stab(n,κ,δ; ) := min{stab(h; ) : H has order n, connectivity κ and minimum degree δ}. Theorem 5 Let κ, δ, N where κ is even, κ δ. Then for each n = p δκ, where p is an arbitrary integer, stab(n,κ,δ; ) δ n + δκn + κ. (6) Proof Let κ, δ be fixed and t = p δ for some integer p. In order to show the upper bound, we construct the following multigraph H(t). LetV i ={v i,1,..., v i,κ/ } and V i ={v i,1,..., v i,κ/ } for i = 0,..., t 1. Then V (H(t)) := V 0 V 0 V 1 V 1... V t 1 V t 1 and ( Vi V ) E(H(t)) = i {v i, j v i+1, j : j = 1,..., κ/}, i = 0,..., t 1, where edges v i, j v i, j, j = 1,..., κ/, have multiplicity δ κ + 1. Therefore, there is a clique built on V i V i, where some edges are multiple (such that G[V i V i ] is δ 1 regular), and there is a perfect matching between V i and V i+1 (i + 1 taen modulo t). It is easy to see that H(t) is δ-regular and has connectivity κ. Furthermore, H(t) =tκ and H(t) = tκδ. We will construct a (H(t); ) stable graph with as few as possible number of edges. Let G (t), 0 t/p, be a graph which arises from H(t) by adding edges {v ip, j v ip+p+1, j,..., v ip, j v ip+p+1, j : j = 1,..., κ/} for all i = 0,..., p t (with indices taen modulo t). We will show, by induction on, that G (t + p) is (H(t); )-stable. This is obvious for = 0. Assume that 1 and the statement is true for G 1 (t + p( 1)). Itis 13

6 37 J Comb Optim (015) 9: suficient to prove that for each x V (G (t + p)), G (t + p) x is (H(t); 1)- stable. Without loss of generality we may assume that x V i V i, i {1,...,p}.It can be seen that [ ] G (t + p) V 0 V 0 V p+1 V p+1... V t+p 1 V t+p 1 G 1 (t + p( 1)). Hence, G 1 (t + p( 1)) is a subgraph of G (t + p) x. Thus, by the induction hypothesis, G (t + p) x is (H(t); 1)-stable. Therefore, G (t + p) is (H(t); )- stable. Finally, G (t + p) = H(t + p) + t+p p Since p = tδ and n = tκ, we obtain κ = (t+p)κδ + t+p κ p G (t + p) = nδ + nδκ + κ. Note that Theorem 5 implies that for = 1 the bound (1) is nearly best possible. Namely, the gap between (1) and (6) depends only on δ, κ and does not depend on n. Acnowledgments Education. The author was partially supported by the Polish Ministry of Science and Higher Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. References Cichacz S, Görlich A, Zwone M, Ża A (011) On (C n ; ) stable graphs. Electron J Comb 18(1):05 Cichacz S, Görlich A, Niodem M, Ża A (01)Alower boundon thesize of (H; 1)-vertex stable graphs. Discrete Math 31: Dude A, Szymańsi A, Zwone M (008) (H, ) stable graphs with minimum size. Discuss Math Graph Theory 8(1): Dude A,Zwone M (009) (H, ) stable bipartite graphs with minimum size. Discuss Math Graph Theory 9: Dude A, Ża A (010) On vertex stability with regard to complete bipartite subgraphs. Discuss Math Graph Theory 30: Fouquet J-L, Thuillier H, Vanherpe J-M, Wojda AP (01) On (K q ; ) vertex stable graphs with minimum size. Discrete Math 31: Fouquet J-L, Thuillier H, Vanherpe J-M, Wojda AP (01) On (K q ; ) stable graphs with small. Electron J Comb 19:50 Franl P, Katona GY (008) Extremal -edge Hamiltonian hypergraphs. Discrete Math 308: Horváth I, Katona GY (011) Extremal P 4 -stable graphs. Discrete Appl Math 16: Ża A (011) On (K q ; )-stable graphs, J. Graph Theory. doi:10.100/jgt