Slim Tree-Cut Width

TL;DR

This paper introduces slim tree-cut width, a new graph parameter that combines desirable structural and algorithmic properties of existing measures, improving upon previous edge-cut based parameters.

Contribution

It defines slim tree-cut width by modifying tree-cut width, demonstrating its structural and algorithmic advantages, and providing characterizations and approximation algorithms.

Findings

Slim tree-cut width satisfies structural and algorithmic requirements.

Provides an alternative spanning-tree characterization.

Includes an approximation algorithm for the parameter.

Abstract

Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an edge-cut based alternative to treewidth in algorithmic aspects. This has led to the very recent introduction of a simple edge-based parameter called edge-cut width [WG 2022], which has precisely the algorithmic applications one would expect from an analogue of treewidth for edge cuts, but does not have the desired structural properties. In this paper, we study a variant of tree-cut width obtained by changing the threshold for so-called thin nodes in tree-cut decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies all the requirements of an edge-cut based analogue of treewidth, both structural and algorithmic, while being less restrictive than edge-cut width. Our results also include an alternative characterization of slim tree-cut width via an easy-to-use spanning-tree decomposition akin to the one used for edge-cut width, a characterization of slim tree-cut width in terms of forbidden immersions as well as approximation algorithm for computing the parameter.