A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth

International audienceThe graph parameter of {\sl 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 {\sl 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 is to be {\sl monotone} (no recontamination occurs) and {\sl 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 [{\sl Dereniowski, Osula, and Rz{\k{a}}{\.{z}}ewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019}\,]. For our algorithm, we enrich the {\sl typical sequence technique} that is able to deal with the connectivity demand. Typical sequences have been introduced in [{\sl 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. 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 so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a $2^{O(k^2)}\cdot n$ time algorithm for the monotone and connected version of the edge search number

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Call number: LD2668 .T4 1964 P31Master of Scienc

Branchwidth of chordal graphs

AbstractThis paper revisits the ‘branchwidth territories’ of Kloks, Kratochvíl and Müller [T. Kloks, J. Kratochvíl, H. Müller, New branchwidth territories, in: 16th Ann. Symp. on Theoretical Aspect of Computer Science, STACS, in: Lecture Notes in Computer Science, vol. 1563, 1999, pp. 173–183] to provide a simpler proof, and a faster algorithm for computing the branchwidth of an interval graph. We also generalize the algorithm to the class of chordal graphs, albeit at the expense of exponential running time. Compliance with the ternary constraint of the branchwidth definition is facilitated by a simple new tool called k-troikas: three sets of size at most k each are a k-troika of set S, if any two have union S. We give a straightforward O(m+n+q2) algorithm, computing branchwidth for an interval graph on m edges, n vertices and q maximal cliques. We also prove a conjecture of Mazoit [F. Mazoit, A general scheme for deciding the branchwidth, Technical Report RR2004-34, LIP — École Normale Supérieure de Lyon, 2004. http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-34.pdf], by showing that branchwidth can be computed in polynomial time for a chordal graph given with a clique tree having a polynomial number of subtrees