Polynomial time recognition of unit circular-arc graphs

We present an efficient algorithm for recognizing unit circular-arc (UCA) graphs, based on a characterization theorem for UCA graphs proved by Tucker in the seventies. Given a proper circular-arc (PCA) graph G, the algorithm starts from a PCA model for G, removes all its circle-covering pairs of arcs and determines whether G is a UCA graph. We also give an O(N) time bound for Tucker’s 3/2-approximation algorithm for coloring circular-arc graphs with N vertices, when a circular-arc model is given.  2004 Elsevier Inc. All rights reserved. * Corresponding author. Fax: (56) (2) 689-7895. E-mail addresses: [email protected] (G. Durán), [email protected] (A. Gravano), [email protected] (R.M. McConnell), [email protected] (J. Spinrad), [email protected] (A. Tucker). 1 Partially supported by FONDECyT Grant 1030498 and Millennium Science Nucleus “Complex Engineering Systems”, Chile and “International Scientific Cooperation Program CONICyT/SETCIP”, Chile–Argentina. 2 Partially supported by UBACyT Grant X127, Argentina. 0196-6774/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgor.2004.08.003 ARTICLE IN PRESS S0196-6774(04)00146-4/FLA AID:1401 Vol.•••(•••) [DTD5] P.2 (1-12) YJAGM:m1 v 1.28 Prn:11/10/2004; 14:22 yjagm1401 by:IS p. 2 2 G. Durán et al. / Journal of Algorithms ••• (••••) •••–•••