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- W2608681412 abstract "Let (mathcal{C}_{H}) be the class of graphs containing some fixed graph H as a minor. We define (mathbf{c}^mathsf{v}_{H}(G)) (resp. (mathbf{c}^mathsf{e}_{H}(G))) as the minimun number of vertices (resp. edges) whose removal from G produces a graph without any subgraph isomorphic to a graph in (mathcal{C}_{H}). Also (mathbf{p}^mathsf{v}_{H}(G)) (resp. (mathbf{p}^mathsf{e}_{H}(G))) is the the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that are isomorphic to some graph in (mathcal{C}_{H}). We denote by (theta _{r}) the graph with two vertices and r parallel edges between them. When (H=theta _{r}), the parameters (mathbf{c}^mathsf{v/e}_{H}) and (mathbf{p}^mathsf{v/e}_{H}) are NP-complete to compute (for sufficiently large r). In this paper we prove a series of combinatorial and algorithmic lemmata that imply that if (mathbf{p}^mathsf{v/e}_{theta _r}(G)le k), then (mathbf{c}^mathsf{v/e}_{theta _r}(G) = O(klog k)). This means that for every r, the class (mathcal{C}_{theta _{r}}) has the vertex/edge Erdős-Posa property. Using the combinatorial ideas from our proofs we introduce a unified approach for the design of an (O(log mathrm{OPT}))-approximation algorithm for (mathbf{c}^mathsf{v}_{theta _{r}}), (mathbf{p}^mathsf{v}_{theta _{r}}), (mathbf{c}^mathsf{e}_{theta _{r}}) and (mathbf{p}^mathsf{e}_{theta _{r}}) that runs in (O(ncdot log (n)cdot m)) steps." @default.
- W2608681412 created "2017-05-05" @default.
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- W2608681412 date "2017-04-28" @default.
- W2608681412 modified "2023-09-24" @default.
- W2608681412 title "An $$O(log mathrm {OPT})$$ O ( log OPT ) -Approximation for Covering and Packing Minor Models of $$theta _r$$ θ r" @default.
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- W2608681412 doi "https://doi.org/10.1007/s00453-017-0313-5" @default.
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