From Pathwidth to Connected Pathwidth

Abstract

It is proven that the connected pathwidth of any graph G is at most 2*pw(G)+1, where pw(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 some graph searching games. One of the advantages of the above bound for connected pathwidth is an inequality $csn(G) <= 2*sn(G)+3$, where $csn(G)$ is the connected search number of a graph $G$ and $sn(G)$ is its search number, which holds for any graph $G$. Moreover, the algorithm presented in this work can be used to convert efficiently a given search strategy using $k$ searchers into a connected one using $2k+3$ searchers and starting at arbitrary homebase.