Certifying algorithms for recognizing proper circular-arc graphs and unit circular-arc graphs

We give two new linear-time algorithms, one for recognizing proper circular-arc graphs and the other for recognizing unit circular-arc graphs. Both algorithms provide either a model for the input graph, or a certificate that proves that such a model does not exist and can be authenticated in O(n) time. No other previous algorithm for each of these two graph classes provides a certificate for its result.