Various results on the toughness of graphs

Abstract

Let G be a graph and let t ≥ 0 be a real number. Then, G is t‐tough if tω(G − S) ≤ |S| for all S ⊆ V(G) with ω(G − S) > 1, where ω(G − S) denotes the number of components of G − S. The toughness of G, denoted by τ(G), is the maximum value of t for which G is t‐tough [taking τ(Kn) = ∞ for all n ≥ 1]. G is minimally t‐tough if τ(G) = t and τ(H) < t for every proper spanning subgraph H of G. We discuss how the toughness of (spanning) subgraphs of G and related graphs depends on τ(G), we give some sufficient degree conditions implying that τ(G) ≥ t, and we study which subdivisions of 2‐connected graphs have minimally 2‐tough squares. © 1999 John Wiley & Sons, Inc. Networks 33: 233–238, 1999

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“…In [53], Broersma, Engbers and Trommel studied the relationship between the toughness of a graph and the toughness of its spanning subgraphs. In particular they proved the following.…”

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