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

TL;DR

This paper extends classical domination and independent set problems to their distance variants, demonstrating their fixed-parameter tractability on graphs with bounded mim-width and identifying complexity boundaries.

Contribution

It introduces distance versions of $(\sigma, ho)$-problems, proves their XP complexity on graphs with bounded mim-width, and shows W[1]-hardness in certain parameterizations.

Findings

Distance problems are XP on graphs with bounded mim-width.

Many classes of graphs have bounded and quickly computable mim-width.

Certain distance problems are W[1]-hard when parameterized by mim-width plus solution size.

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.