A LINEAR FIXED PARAMETER TRACTABLE ALGORITHM FOR CONNECTED PATHWIDTH

Abstract

The graph parameter of pathwidth can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of node search, where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one. Two desired characteristics for a cleaning strategy are to be monotone (no recontamination occurs) and connected (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called connected pathwidth. We prove that connected pathwidth is fixed parameter tractable; in particular we design a 2O(k2)\cdot n time algorithm that checks whether the connected pathwidth of G is at most k. This resolves an open question by Dereniowski, Osula, and Rz\k a\.zewski [Theoret. Comput. Sci., 794 (2019), pp. 85-100]. For our algorithm, we enrich the typical sequence technique that is able to deal with the connectivity demand. Typical sequences have been introduced by Bodlaender and Kloks [J. Algorithms, 21 (1996), pp. 358-402] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a ``global'' demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a 2O(k2) \cdot n time algorithm for the monotone and connected version of the edge search number.