Mim-Width III. Graph powers and generalized distance domination problems
Peer reviewed, Journal article
Published version
![Thumbnail](/bora-xmlui/bitstream/handle/1956/22107/00000-mim-iii.pdf.jpg?sequence=6&isAllowed=y)
View/ Open
Date
2019Metadata
Show full item recordCollections
Original version
https://doi.org/10.1016/j.tcs.2019.09.012Abstract
We generalize 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 in XP parameterized by the structural parameter mim-width, and hence polynomial-time solvable 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. We obtain these results by showing that taking any power of a graph never increases its mim-width by more than a factor of two. To supplement these findings, we show that many classes of (σ,ρ) problems are W[1]-hard parameterized by mimwidth + solution size. We show that powers of graphs of tree-width w − 1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1.