Induced Disjoint Paths in Circular-Arc Graphs in Linear Time

The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (si,ti) contains paths P1,…,Pk such that Pi connects si and ti for i=1,…,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤

Induced disjoint paths in circular-arc graphs in linear time

The Induced Disjoint Paths problem is to test whether an graph G on n vertices with k distinct pairs of vertices (si,ti) contains paths P1,…,Pk such that Pi connects si and ti for i=1,…,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤

Subexponential Algorithms for Partial Cover Problems

Partial Cover problems are optimization versions of fundamental and well studied problems like {\sc Vertex Cover} and {\sc Dominating Set}. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number $k$ of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by $k$. It was recently shown by Amini et. al. [{\em FSTTCS 08}\,] that {\sc Partial Vertex Cover} and {\sc Partial Dominating Set} are fixed parameter tractable on large classes of sparse graphs, namely $H$.minor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2^{\cO(k)}n^{\cO(1)}.publishedVersio

The (theta, wheel)-free graphs Part IV: Induced paths and cycles

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins. In this paper we use this decomposition theorem to solve several problems related to finding induced paths and cycles in our class

Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width

We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give n^O(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H-Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Per- mutation and Circular Permutation graphs, Convex graphs, k-Trapezoid, Circular k-Trapezoid, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k

Few induced disjoint paths for H-free graphs

Paths P1,…,Pk in a graph G=(V,E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices. For a fixed integer k, the k-INDUCED DISJOINT PATHS problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi starts from si and ends at ti. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-INDUCED DISJOINT PATHS is NP-complete. We prove new complexity results for k-INDUCED DISJOINT PATHS if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for INDUCED DISJOINT PATHS, the variant where k is part of the input