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arXiv · 1705.10570

The complexity of recognizing minimally tough graphs

Gyula Y Katona·István Kovács·Kitti Varga
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Abstract

A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. The complexity class DP is the set of all languages that can be expressed as the intersection of a language in NP and a language in coNP. In this paper, we prove that recognizing minimally $t$-tough graphs is DP-complete for any positive rational number $t$. We introduce a new notion called weighted toughness, which has a key role in our proof.

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Gyula Y Katona, István Kovács, Kitti Varga. 2017-05-30. The complexity of recognizing minimally tough graphs. https://doi.org/10.1016/j.dam.2021.01.022

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