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Seventh Cologne-Twente Workshop on Graphs and Combinatorial Optimization Seventh Cologne-Twente Workshop on Graphs and Combinatorial Optimization Scientific Program Gargnano, Italy May 13th to 15th, 2008 2 INDEX: List of Corresponding Authors: page 197 List of Participants: page 199 Contributed Papers: Track A Graph theory (I) (A1) page 1 Coloring 1 (A2) page 31 Polyhedra and formulations (A3) page 57 Graph theory (II) (A4) page 87 Coloring 2 (A5) page 117 Graph theory (III) (A6) page 143 Mathematical programming (A7) page 173 Track B Network design (B1) page 15 Graph optimization (I) (B2) page 43 On-line optimization and uncertainty (B3) page 71 Graph optimization (II) (B4) page 103 Routing (B5) page 131 Column generation (B6) page 157 Location (B7) page 185 4 Graph theory (I) Room A, Session 1 Tuesday 13, 10:30-12:00 1 Improving the gap of Erdos-Pósa˝ property for minor-closed graph classes 1 a,2 a,2 b,3, Fedor V. Fomin Saket Saurabh Dimitrios M. Thilikos ∗ aDepartment of Informatics University of Bergen PO Box 7800, 5020 Bergen, Norway bDepartment of Mathematics, University of Athens, Panepistimioupolis, GR-15784 Athens, Greece. Abstract Let and be graph classes. We say that has the Erdos-Pósa˝ property for if for anyH graph GG , the minimum vertex coveringH of all -subgraphs of G is boundedG by a function f ∈of G the maximum packing of -subgraphsH in G (by -subgraph of G we mean any subgraph of G that belongs to ).H In his monograph “GraphH Theory”, R. Diestel proves that if is the class of all graphsH that can be contracted to a fixed planar graph H, then hasH the Erdös-Pósa property for the class of all graphs (with an exponential boundingH function). In this note, we give an alternative proof of this result with a better (still exponential) bounding function. Our proof, for the case when is some non-trivial minor-closed graph class, yields a low degree polynomial bounding functioG n f. In particular f(k)= O(k1.5). Key words: Erdos-Pósa˝ property, treewidth, graph packing, graph covering 1. Introduction Given a graph G we denote by V (G) and E(G) its vertex and edge set respectively. A graph class is called non-trivial if it does not contain all graphs. We use the notation H G to denote that H is a subgraph of G. We say that a graph G is a -subgraph⊆ of a graph G′ if G G′ and G . G ⊆ ∈ G 1 This research has been done while the first two authors were visiting the Department of Mathematics of the National and Kapodistrian University of Athens during October 2007. 2 Supported by the Research Council of Norway. 3 Supported by the Project “Kapodistrias” (AΠ 736/24.3.2006) of the National and Kapodistrian University of Athens (project code: 70/4/8757). CTW08 - Università degli Studi di Milano, Gargnano (Italy), May 13-15 2008 Let be a class of graphs. Given a graph G, we define the covering number of G withH respect to the class as H cover (G) = min k S V (G) H * G S . H { | ∃ ⊆ ∀H∈H 6 − } In other words, coverH(G) k if there is a set of at most k vertices meeting any -subgraph of G. We also define≤ the packing number of G with respect to the class H as H pack (G) = max k a partition V ,...,V of V (G) H { | ∃ 1 k such that H G[V ] . ∀i∈{1,...,k}∃H∈H ⊆ i } Less formally, pack (G) k if G contains k vertex-disjoint -subgraphs. H ≥ H A graph class satisfies the Erdös-Pósa property for some graph class if there is a function f H(depending only on and ) such that, for any graph G G , H G ∈ G pack (G) cover (G) f(pack (G)). (1) H ≤ H ≤ H We say that a graph G can be contracted to H if H can be obtained from G after a series of edge contractions (the contraction of an edge e = (u, v) in G results ′ in a graph G , in which u and v are replaced by a new vertex ve and in which for ′ every neighbour w of u or v in G, there is an edge (w,ve) in G ). We say that H is a minor of G if some subgraph of G can be contracted to H. We say that a graph class is minor-closed if any minor of a graph in is again a member of . We denoteG by (H) the class of graphs that can be contractedG to H. G Theorem 1M (Corollary 12.3.10 and Exercise 39 in [2]) (H) satisfies the Erdös- Pósa property for all graphs if and only if H is a connectedM planar graph. According to the proof of Theorem 1, the bounding function f(k) of Relation (1) is exponential in k. Given a connected planar graph H and a graph class we denote G by f : N N the optimal upper bounding function with the property that, for H,G → any graph G , Relation (1) holds when = (H) (we know that fH,G exists because of Theorem∈ G 1). H M For instance, according to the classic result of Erdös and Pósa [3] if H = K3 and contains all the graphs, then f (k)= O(k log k). However, if we restrict to G H,G · G be a class of planar graphs, then fH,G(k) = O(k) [4]. In this note, we conjecture that this last fact extends when H is any planar graph and when is any non-trivial minor-closed graph class. G Conjecture 2 If is a non-trivial minor-closed graph class, then fH,G is a linear function for any planarG graph H. The purpose of this short note is to prove that the above conjecture holds if we ask for a polynomial bound for fH,G, thereby sharpening the result of Diestel in [2]. 3 2. The main result and the proof Our main result is the following: Theorem 3 Let H be for some planar graph and let be the class of graphs that are contractible to H. Let also be a non-trivial minor-closedH graph class. Then G there is a constant cG,H depending only on and H such that for every graph G , it holds that G ∈ G pack (G) cover (G) c (pack (G))3/2. H ≤ H ≤ G,H · H The proof of Theorem 3 will be the immediate consequence of Lemmata 1 and 2 below. Before we state and prove them, we need first to define the notions of tree decomposition and treewidth. Let G be a graph. A tree decomposition of G is a pair (T, = Xt t∈V (T )) where the following conditions hold. X { } = V (G) • ∪u∈V (T ) e∈E(G) t∈V (T ) : e Xt • ∀ ∃T [ t v X⊆ ] is connected. • ∀v∈V (G) { | ∈ t} The width of a tree decomposition is maxt∈V (T ) X 1 and the treewidth of G is the minimum width over all the tree decompositions| | of− G. Our first observation is the following. Lemma 1 Let H be a connected planar graph and let = (H). Let also be H M G a non-trivial minor closed graph class. Then, there is a constant cG,H , depending only on and H such that for any graph G, tw(G) c (pack (G))1/2. G ≤ G,H · H Proof. Let k = packM(H)(G). During this proof, for any positive integer t we will denote by Γ the (t t)-grid. Let t × c = min r H is a minor of the (r r)-grid H { | × } and notice that if m = k1/2 + 1, then pack (Γ ) > k. We conclude that G ⌈ ⌉ M(H) m·cH does not contain Γm·cH as a minor. From the main result in [1], there is a constant c depending only on such that tw(G) c m c and the lemma follows. G G ≤ G · · H For the proof of the next Lemma, we will enhance the definition of a tree decom- position (T, ) as follows: T is a tree rooted on some node r where Xr = , each of its nodes haveX at most two children and can be one of the following ∅ ′ (1) Introduce node: a node t that has only one child t where Xt Xt′ and such that t′ is not an introuce node. ⊃ ′ (2) Forget node: a node t that has only one child t where Xt Xt′ and such that t′ is not a forget node. ⊂ (3) Join node: a node t with two children t1 and t2 such that Xt = Xt1 = Xt2 . 4 (4) Base node: a node t that is a leaf of t, is different than the root and X = . t ∅ Notice that, according the the above definitions, the root r of T is either a forget or a join node. It is easy to see that any tree decomposition can be transformed to one with the above requirements while maintaining the same width. From now one when we refer to a tree decomposition (T, ) we will presume the above require- ments. X Given a tree decomposition (T, ) and some node t of T , we define as T the X t subtree of T rooted on t. Clearly, if r is the root of T , it holds that Tr = T . We also − define Gt = G[ s∈V (Tt)Xs] and Gt = Gt Xt. Lemma 2 Let H∪ be a connected planar graph− and let = (H). Then for any graph G, it holds that cover (G) (tw(G)+1) packH(G).M H ≤ · H Proof.
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