Generalized distance domination problems and their complexity on graphs of bounded mim-width

Lars Jaffke; O-joung Kwon; Torstein J. F. Strømme; Jan Arne Telle

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arXiv · 1803.03514

Generalized distance domination problems and their complexity on graphs of bounded mim-width

Lars Jaffke·O-joung Kwon·Torstein J. F. Strømme·Jan Arne Telle

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Abstract

We generalize the family of $(\sigma, \rho)$-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-$r$ dominating set and distance-$r$ independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as $k$-trapezoid graphs, Dilworth $k$-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, $k$-polygon graphs, circular arc graphs, complements of $d$-degenerate graphs, and $H$-graphs if given an $H$-representation. To supplement these findings, we show that many classes of (distance) $(\sigma, \rho)$-problems are W[1]-hard parameterized by mim-width + solution size.

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BibTeX RIS

Lars Jaffke, O-joung Kwon, Torstein J. F. Strømme, Jan Arne Telle. 2018-03-09. Generalized distance domination problems and their complexity on graphs of bounded mim-width. https://arxiv.org/abs/1803.03514

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