A simple graph \(G\) is said to be \(r\)-regular if the degree of very vertex of \(G\) is \(r\), and \(G\) is called endo-regular (respectively, endo-orthodox, endo-completely-regular) if its monoid of all endomorphisms is regular (respectively, orthodox, completely regular).
The authors of the article under review show that the complements of the cycles \(C_n\) (\(n\geq3\)) are endo-completely-regular, which improves the results obtained by M. W. Li [Discrete Math. 265, No. 1–3, 105–118 (2003; Zbl 1015.05031)]. It is also proved that the \((n-3)\)-regular graph of order \(n\) is endo-regular, and the conditions under which the \((n-3)\)-regular graph of order \(n\) is endo-completely-regular and endo-orthodox are given, respectively.