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We generalize the family of $(\sigma, \rho)$-problems and locally checkablevertex partition problems to their distance versions, which naturally captureswell-known problems such as distance-$r$ dominating set and distance-$r$independent set. We show that these distance problems are XP parameterized bythe structural parameter mim-width, and hence polynomial on graph classes wheremim-width is bounded and quickly computable, such as $k$-trapezoid graphs,Dilworth $k$-graphs, (circular) permutation graphs, interval graphs and theircomplements, convex graphs and their complements, $k$-polygon graphs, circulararc 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.

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