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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.

MSC:
 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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