A Note on the Toughness of Certain Graphs

The toughness \(t(G)\) of a noncomplete graph \(G\) is defined as

\[t(G) = \min{\left\{\frac{|S|}{\omega(G-S)} \mid S \subset V(G), \omega(G-S) \geq 2\right\}}\]

where \(\omega(G-S)\) is the number of components of \(G-S\). We also define \(t(K_n) = +\infty\) for every \(n\).

In this article, we discuss the toughness of the endline graph of a graph and the middle graph of a graph.