Unit Circular-Arc Graph Representations and Feasible Circulations

In a recent paper, Durán, Gravano, McConnell, Spinrad and Tucker described an algorithm of complexity O(n2) for recognizing whether a graph G with n vertices and m edges is a unit circular-arc (UCA) graph. Furthermore the following open questions were posed in the above paper: (i) Is it possible to construct a UCA model for G in polynomial time?; (ii) Is it possible to construct a UCA model, whose extremes of the arcs correspond to integers of polynomial size?; (iii) If (ii) is true, could such a model be constructed in polynomial time? In the present paper, we describe a characterization of UCA graphs, based on network circulations. The characterization leads to a different recognition algorithm and to answering these questions, in the affirmative. We construct a UCA model whose extremes of the arcs correspond to integers of size O(n). The proposed algorithms, for recognizing UCA graphs and constructing UCA models, have complexities O(n+ m). Furthermore, the complexities reduce to O(n), if a proper circular-arc (PCA) model of G is already given as the input, provided the extremes of the arcs are ordered. We remark that a PCA model of G can be constructed in O(n+m) time, using the algorithm by Deng, Hell and Huang. Finally, we also describe a linear time algorithm for finding feasible circulations in networks with non negative lower capacities and unbounded upper capacities. Such an algorithm is employed in the model construction for UCA graphs.