
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…
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