Abstract
We consider tolerance graphs as defined by Golumbic, Monma, and Trotter (1984). These authors proposed a conjecture which can be stated as follows: If H is a comparability graph, then its complement H̄ is a tolerance graph if and only if it is a bounded tolerance graph. A result which supports this conjecture was obtained by one of us in his Diploma-Thesis (Hennig, 1988) where it was shown that, for a tree T , the following three conditions are equivalent: (i) T̄ is a tolerance graph, (ii) T̄ is a bounded tolerance graph, (iii) T does not contain T 3 as a subtree, where T 3 denotes the tree which consists of three paths of length 3 starting at a common vertex. It is the purpose of the present note to give a short proof of this result.
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