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PTAS for \(\mathcal{H}\)-free node deletion problems in disk graphs. (English) Zbl 1382.05037
Summary: For a set \(\mathcal{H}\) of graphs, a graph \(G\) is \(\mathcal{H}\)-free if \(G\) does not contain any subgraph isomorphic to some graph in \(\mathcal{H}\). In this paper, we study the minimum \(\mathcal{H}\)-free node deletion problem (\(\text{Min}\mathcal{H}\text{FND}\)) and the maximum \(\mathcal{H}\)-free node set problem (\(\text{Max}\mathcal{H}\text{FND}\)), which include a lot of extensively-studied problems such as the minimum \(k\)-path vertex cover problem, the dissociation number problem, and the minimum degree bounded node deletion problem. For a large class of \(\mathcal{H}\), PTASs are given for \(\text{Min}\mathcal{H}\text{FND}\) and \(\text{Max}\mathcal{H}\text{FND}\) on disk graphs whose heterogeneity is bounded by a constant, where the heterogeneity of a disk graph is the ratio of the maximum radius to the minimum radius of disks.

05C35 Extremal problems in graph theory
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C40 Connectivity
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