# The complexity of recognizing minimally tough graphs

## TL;DR

This paper proves that recognizing minimally t-tough graphs is DP-complete for any positive rational t, introducing weighted toughness as a key concept in the proof.

## Contribution

It establishes the DP-completeness of recognizing minimally t-tough graphs for all positive rational t and introduces weighted toughness to aid the proof.

## Key findings

- Recognition problem is DP-complete for all positive rational t.
- Introduces weighted toughness as a new concept.
- Provides complexity classification for minimally t-tough graphs.

## 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.