An edge variant of the Erdős–Pósa property

Abstract

For every r ∈ N, we denote by θ r the multigraph with two vertices and r parallel edges. Given a graph G, we say that a subgraph H of G is a model of θ r in G if H contains θ r as a contraction. We prove that the following edge variant of the Erdős-Pósa property holds for every r 2: if G is a graph and k is a positive integer, then either G contains a packing of k mutually edge-disjoint models of θ r , or it contains a set S of f r (k) edges such that G \ S has no θ r -model, for both f r (k) = O(k 2 r 3 polylog kr) and f r (k) = O(k 4 r 2 polylog kr). Erdős-Pósa property with gap O(k · log k). Given a graph J, we denote by M(J) the set of all graphs containing J as a contraction. Robertson and Seymour proved the following proposition, which in particular can be seen as an extension of the Erdős-Pósa Theorem. Proposition 1. Let J be a graph. The class M(J) satisfies the Erdős-Pósa property if and only if J is planar. A proof of Proposition 1 appeared for the first time in [18]. Another proof can be found in Diestel's monograph [4, Corollary 12.4.10 and Exercise 40 of Chapter 12]. In view of Proposition 1, it is natural to try to derive good estimations of the gap function f M(J) in the case where J is a planar graph. In this direction, the recent breakthrough results of Chekuri and Chuzhoy imply that f M(J) (k) = k · polylog k [2] when J is a planar graph and, even more, that f M(J) = (k + |V (J)|) O(1) [3]. Before this, the best known estimation of the gap for planar graphs was exponential, namely f M(J) (k) = 2 O(k log k) , and could be deduced from [14] using the proof of [18]. Moreover, some improved polynomial gaps have been proven for particular instantiations of the graph J (see [9,15,16,7,8,6]). Another direction is to add restrictions on the graphs G that we consider, which usually allows to optimize the gap f M(J) . In this direction, it is known that f M(J) = O(k) in the case where graphs are restricted to some non-trivial minor-closed class [10].We consider the edge counterpart of the Erdős-Pósa property, where packings are edgedisjoint (instead of vertex-disjoint) and coverings contain edges instead of vertices. We say that a graph class G satisfies the edge variant of the Erdős-Pósa property if there exists a function f G such that, for every graph G and every positive integer k, either G contains k mutually edge-disjoint subgraphs, each isomorphic to a graph in G, or it contains a set X of f G (k) edges meeting every subgraph of G that is isomorphic to a graph in G. Recently, the edge variant of the Erdős-Pósa property was proved in [12] for 4-edge-connected graphs in the case where G contains all odd cycles. J (G) for the minimum size of a subset S ⊆ E(G) (called J-edge-hitting set) that meets the edge sets of all models of J in G. Obviously, for every two graphs G and J, the following inequality holds:A graph J is said to satisfy the (vertex-)Erdős-Pósa property for minors (vertex-Erdős-Pósa property for short) if there is a function f J : N → N, called vertex-Erdős-Pósa gap of J, such that for e...

An $$O(\log \mathrm {OPT})$$ O ( log OPT ) -Approximation for Covering and Packing Minor Models of $$\theta _r$$ θ r

“…This verifies the optimal combinatorial bound for the case where x = v [14] and optimally improves the previous bound in [31] for the case where x = e. Based on the proof of Theorem 1, we prove the following algorithmic result.…”

“…As mentioned in [31], the planarity of a graph H is a necessary condition for the minor models of H to have the edge-Erdős-Pósa property. However, little is known on which planar graphs have this property and with which gap.…”

“…Moreover, as proved in [31], the class C θ r has the e-EP-property with the (non-optimal) gap f (k) = O(k 2 · log O(1) k). Interestingly, not much more is known on the graphs H for which C H has the e-EP-property and is tempting to conjecture that the planarity of H provides again the right dichotomy.…”

K4-expansions have the edge-Erdős-Pósa property

“…What is a lower bound on the size of the edge hitting set for K 4 -subdivisions? Fiorini et al note in the introduction of [10] that there are graphs G on n vertices with treewidth Ω(n) and girth Ω(log n) and Raymond et al [17] mention that these graphs even can be chosen cubic. Every K 4 -subdivision in such a graph G contains at least Ω(log n) vertices as it contains a cycle and the girth of G is Ω(log n).…”

“…Compared to the ordinary (vertex-)Erdős-Pósa property there are only a few classes known to have the edge-Erdős-Pósa property: cycles have the edgeproperty (see, for instance, [7,Exercise 9.5]), as do long cycles [2] and as do θ r -expansions (see Raymond, Sau and Thilikos [17]), where θ r is the multigraph consisting of r parallel edges.…”