Circular-arc graph recognition and related problems

Circular-arc graphs have a rich combinatorial structure and it has been suggested that this is why complicated ad hoc methods, instead of general principles, are the basis of almost all results in the theory of circular-arc graphs. We establish a relationship between circular-arc graphs and chordal bipartite graphs which captures some of the complex structure of circular-arc graphs and provides a more unified approach to designing efficient algorithms for solving problems on circular-arc graphs. The problem of recognizing circular-arc graphs was initially conjectured to be NP-Complete and the question remained open for about sixteen years. In 1980, Alan Tucker proved the conjecture false by devising an $O(n\sp3)$ time algorithm. We exploit the relationship between circular-arc graphs and chordal bipartite graphs to achieve an $O(n\sp2)$ time circular-arc graph recognition algorithm which is simpler than Tucker's algorithm. We retain the overall structure of Tucker's algorithm, but introduce new techniques in each of the parts. The algorithm employs a new $O(n\sp2)$ time method for computing neighborhood containment relations in circular-arc graphs, and reductions to the recognition problem on two clique coverable circular-arc graphs. As further applications of the relationship between circular-arc graphs and chordal bipartite graphs, we present an $O(n\sp2)$ isomorphism test for two clique coverable circular-arc graphs, a new $O(n\sp2)$ time algorithm to compute neighborhood intersection relations in circular-arc graphs, and we improve the time complexity of generating all maximum cliques of a circular-arc graph.