Discs in unbreakable graphs (Q1900520)

A graph \(G\) is said to be unbreakable if it has at least three vertices and neither the graph nor its complement has a star cutset (i.e., a non- empty subset \(C\) of the vertices of \(G\), for which \(G- C\) is disconnected and some vertex of \(C\) is adjacent to all other vertices of \(C\)). In this paper it is shown that every vertex in an unbreakable graph is in a disc, where a disc is a chordless cycle, or the complement of a chordless cycle, with at least five vertices. A corollary is that every vertex in a minimal imperfect graph is in a disc.