A graph is a unit interval graph (respectively, an \(\tilde{n}\)-graph) if we can assign to each vertex an open interval of unit length (respectively, a set of \(n\) consecutive integers) so that edges correspond to pairs of intervals (respectively, of sets) that overlap. Sakai [14] and Troxell [18] provide a linear time algorithm to find the smallest integer \(n\) so that a unit interval graph is an \(\mathbb{A}\)-graph, for the particular case of reduced connected graphs with chromatic number \(3\). This work shows how to obtain such smallest \(n\) for arbitrary graphs, by establishing a relationship with the work by Bogart and Stellpflug [1] in the theory of semiorders.
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