A sufficient condition for boundedness of tolerance graphs

Golumbic, Monma, and Trotter showed that every tolerance graph for which no vertex neighborhood is contained in another vertex neighborhood is a bounded tolerance graph. We strengthen this result by weakening the neighborhood condition. In this way, more tolerance graphs can be recognized as bounded. Our argument relies on a variation of the concept of “assertive vertices”.

► We strengthen a result concerning a neighborhood condition on bounded tolerance graphs.
► More tolerance graphs can be recognized as bounded.
► Bounded tolerance graphs are characterized using a variation of the notion of assertive vertices.