From Pathwidth to Connected Pathwidth

TL;DR

This paper proves an upper bound on the connected pathwidth of graphs based on their pathwidth, provides an efficient algorithm for constructing connected path decompositions, and explores implications for graph search strategies.

Contribution

It introduces a constructive method to convert any path decomposition into a connected one with at most double the width plus one, along with an efficient algorithm and applications to search strategies.

Findings

Connected pathwidth is at most twice the pathwidth plus one.

An efficient algorithm computes connected path decompositions from given path decompositions.

The method improves bounds on the connected search number relative to the search number.

Abstract

It is proven that the connected pathwidth of any graph $G$ is at most $2\cdot\pw(G)+1$, 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 the search number of a graph. One of the advantages of the above bound for connected pathwidth is an inequality $\csn(G)\leq 2\sn(G)+3$, where $\csn(G)$ and $\sn(G)$ 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.