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

We generalise the family of (σ, ρ)-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, H-graphs and complements to d-degenerate graphs. To supplement these findings, we show that many classes of (distance) (σ, ρ)-problems are W[1]-hard parameterized by mim-width + solution size.