Finding small-width connected path decompositions in polynomial time

TL;DR

This paper proves that for any fixed width k, the connected pathwidth of a graph can be computed efficiently in polynomial time, resolving an open problem in graph theory and related search games.

Contribution

It establishes a polynomial-time algorithm for computing small-width connected path decompositions for any fixed k, solving an open problem in the field.

Findings

Connected pathwidth can be computed in polynomial time for fixed k.

Addresses an open question from GRASTA 2017.

Connects connected pathwidth to the connected node search game.

Abstract

A connected path decomposition of a simple graph $G$ is a path decomposition $(X_1,\ldots,X_l)$ such that the subgraph of $G$ induced by $X_1\cup\cdots\cup X_i$ is connected for each $i\in\{1,\ldots,l\}$. The connected pathwidth of $G$ is then the minimum width over all connected path decompositions of $G$. We prove that for each fixed $k$, the connected pathwidth of any input graph can be computed in polynomial-time. This answers an open question raised by Fedor V. Fomin during the GRASTA 2017 workshop, since connected pathwidth is equivalent to the connected (monotone) node search game.