How tough is toughness?

The concept of toughness was introduced by Chvátal [34] more than forty years ago. Toughness resembles vertex connectivity, but is different in the sense that it takes into account what the effect of deleting a vertex cut is on the number of resulting components. As we will see, this difference has major consequences in terms of computational complexity and on the implications with respect to cycle structure, in particular the existence of Hamilton cycles and k-factors

The Complexity of Recognizing Tough Cubic Graphs

We show that it is NP-hard to determine if a cubic graph G is 1-tough. We then use this result to show that for any integer t # 1, it is NP-hard to determine if a 3 t-regular graph is t-tough. We conclude with some remarks concerning the complexity of recognizing certain subclasses of tough graphs. Keywords : toughness, cubic graphs, NP-completeness AMS Subject Classifications (1991) : 68R10, 05C38 # Supported in part by NATO Collaborative Research Grant CRG 921251. + Supported by a grant from the Natural Sciences and Engineering Council of Canada. # Current address : Centre for Discrete and Applicable Mathematics, Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, England, U.K. Supported in part by the National Science Foundation under Grant DMS9206991. 1 1 Introduction We begin with a few definitions and some notation. A good reference for any undefined terms is [7]. We consider only undirected graphs with no loops or multiple edges...