Covering and packing pumpkin models

Abstract : Let θr (the r-pumpkin) be the multi-graph containing two vertices and r parallel edges between them. We say that a graph is a a θr-model if it can be transformed into θr after a (possibly empty) sequence of contractions. We prove that there is a function g : N → N such that, for every two positive integers k and q, if G is a Kq-minor-free graph, then either G contains a set of k vertex-disjoint subgraphs (a θr-model-vertex-packing) each isomorphic to a θr-model or a set of g(r)·log q ·k vertices (a θr-model-vertex-cover) meeting all subgraphs of G that are isomorphic to a θr-model. Our results imply a O(log OP T)-approximation for the maximum (minimum) size of a θr-model packing (θr-model covering) of a graph G.
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Dimitris Chatzidimitriou, Jean-Florent Raymond, Ignasi Sau, Dimitrios M. Thilikos. Covering and packing pumpkin models. ICGT: International Colloquium on Graph Theory and combinatorics, Jun 2014, Grenoble, France. ⟨lirmm-01083652⟩

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