Let \(G\) be a graph of order \(n\) and \( X\) a given vertex subset of \(G\). Define the parameters:
\(\alpha(V) = \max\{|S| \mid S\}\) is an independent set of vertices of the subgraph \(G(X)\) in \(G\) induced by \(X\)
and
\(\sigma_k(X) = \min\{|\Sigma_{i=1}^{k}d(x_i)| \mid \{x_1,x_2,\ldots,x_k\} \}\) is an independent vertex set in \( G[X]\)
A cycle \(C\) of \(G\) is called \(X\)-longest if no cycle of \(G\) contains more vertices of \(X\) than \(C\). A cycle \(C’\) of \(G\) is called \(X\)-dominating if all neighbors of each vertex of \(X\setminus V(C)\) are on \(C\). In particular, \(G\) is \(X\)-eyclable if \(G\) has an \(X\)-cycle, i.e., a cycle containing all vertices of \(X\). Our main result is as follows:
If \(G\) is \(1\)-tough and \(\sigma_3(X) \geq n\), then \(G\) has an \(X\)-longest cycle \(C\) such that \(C\) is an \(X\)-dominating cycle and \(|V(C) \cap X| \geq \min\{|X|. |X| + \frac{1}{3}\sigma_3(X) – \sigma(X)\}\), which extends the well-known results of D. Bauer et al. [2] in terms of \(X\)-cyclability. Finally, if \(G\) is \(2\)-tough and \(\sigma_3(X) \geq n\), then \(G\) is \(X\)-cyelable.