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 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 is 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 {\em connected pathwidth}. We prove that connected pathwidth is fixed parameter tractable, in particular we design a $2^{O(k^2)}\cdot n$ time algorithm that checks whether the connected pathwidth of $G$ is at most $k.$ This resolves an open question by [Dereniowski, Osula, and Rz{ą}{z}ewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85-100, 2019]. For our algorithm, we enrich the typical sequence technique that is able to deal with the connectivity demand. Typical sequences have been introduced in [Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358-402, 1996] for the design of linear parameterized algorithms for treewidth and pathwidth. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well.
Highlights
Pathwidth can be seen as a measure of the topological resemblance of a graph to a path
On the positive side, connected pathwidth is closed under contractions, i.e, its value does not increase when we contract edges and, the yes-instances of the problem have bounded pathwidth, they have bounded treewidth
To compress a (Gi, P)-encoding sequence S, we identify a subset bp(S) of indexes, called breakpoints, such that j ∈ bp(S) if bd(sj−1) = bd(sj) or cc(sj−1) = cc(sj) or j is an index belonging to a typical sequence of the integer sequence val(sb), . . . , val(sc−1) where b, c ∈ [ ] are consecutive type-1 or 2- breakpoints
Summary
On the positive side, connected pathwidth is closed under contractions (see e.g., [1]), i.e, its value does not increase when we contract edges and, the yes-instances of the problem have bounded pathwidth, they have bounded treewidth Based on these observations, the existence of a parameterized algorithm would be implied if we can prove that, for any k, the set Zk of contraction-minimal graphs with connected pathwidth more than k is finite: as contraction containment can be expressed. While this is a “global property”, it appears that its evolution with respect to the bags of the decomposition can be controlled by the second component of our encoding and this is done in terms of a sequence of a gradually coarsening partitions This establishes a dynamic programming framework that can potentially be applied on the connected versions of most of the parameters where the typical sequence technique was used so far. These two reductions imply that the result of Theorem 1 holds for mcns and mces, i.e., the search numbers for the monotone and connected versions of both node and edge searching
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