Minors in graphs of large θr-girth

For every r∈N, let θr denote the graph with two vertices and r parallel edges. The θr-girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to θr. This notion generalizes the usual concept of girth which corresponds to the case r=2. In Kühn and Osthus (2003), Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of θr-girth and we show that the minimum degree can be replaced by some connectivity measurement. As an application of our results, we prove that, for every fixed r, graphs excluding as a minor the disjoint union of kθr’s have treewidth O(k⋅logk). © 2017 Elsevier Lt

An O(log OPT)-Approximation for Covering and Packing Minor Models of θr

Given two graphs G and H, we define v- coverH(G) (resp. e- coverH(G)) as the minimum number of vertices (resp. edges) whose removal from G produces a graph without any minor isomorphic to H. Also v- packH(G) (resp. e- packH(G)) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that contain a minor isomorphic to H. We denote by θr the graph with two vertices and r parallel edges between them. When H= θr, the parameters v- coverH, e- coverH, v- packH, and e- packH are NP-hard to compute (for sufficiently big values of r). Drawing upon combinatorial results in Chatzidimitriou et al. (Minors in graphs of large θr-girth, 2015, arXiv:1510.03041), we give an algorithmic proof that if v-packθr(G)≤k, then v-coverθr(G)=O(klogk), and similarly for e-packθr and e-coverθr. In other words, the class of graphs containing θr as a minor has the vertex/edge Erdős–Pósa property, for every positive integer r. Using the algorithmic machinery of our proofs we introduce a unified approach for the design of an O(log OPT) -approximation algorithm for v-packθr, v-coverθr, e-packθr, and e-coverθr that runs in O(n· log (n) · m) steps. Also, we derive several new Erdős–Pósa-type results from the techniques that we introduce. © 2017, The Author(s)