2,239 research outputs found
A Unified Polynomial-Time Algorithm for Feedback Vertex Set on Graphs of Bounded Mim-Width
We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an n^{O(w)}-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs for fixed k. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013].
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 extensively 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. Given a tree decomposition of width w-1, a path decomposition of width w, or a clique-width w-expression of a graph G, one can for any value of k find a mim-width decomposition of its k-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the k-power in time n^{O(w)}.
In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width 1, which further hints at the expressive power of the mim-width parameter
Mim-Width III. Graph powers and generalized distance domination problems
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.publishedVersio
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
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