# From Pathwidth to Connected Pathwidth

Dariusz Dereniowski

Arxiv ID: 1007.1269•Last updated: 3/5/2021

It is proven that the connected pathwidth of any graph G is at most 2·(G)+1, where (G) is the pathwidth of G. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width k computes a connected path decomposition of width at most 2k+1. The running time of the algorithm is O(dk^2), where d is the number of `bags' in the input path decomposition. The motivation for studying connected path decompositions comes from the connection between the pathwidth and the search number of a graph. One of the advantages of the above bound for connected pathwidth is an inequality (G)≤ 2(G)+3, where (G) and (G) are the connected search number and the search number of G. Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a (monotone) connected one using 2k+3 searchers and starting at an arbitrary homebase.

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