$O(M\cdot N)$ Algorithms for the Recognition and Isomorphism Problems on Circular-Arc Graphs

Abstract

Circular-arc graphs have a rich combinatorial structure. The circular endpoint sequence of arcs in a model for a circular-arc graph is usually far from unique. We present a natural restriction on these models to make it meaningful to define the unique representations for circular-arc graphs. We characterize those circular-arc graphs which have unique restricted models and give an $O(m \cdot n)$ algorithm for recognizing circular-arc graphs. We think a more careful implementation could reduce the complexity to $O(n^{2})$. Our approach is to reduce the recognition problem of circular-arc graphs to that of circle graphs. This approach has the following advantages: it is conceptually simpler than Tucker’s $O(n^{3})$ recognition algorithm: it exploits the similarity between circle graphs and circular-arc graphs in a natural fashion: it yields an isomorphism algorithm. A main contribution of this result is an illustration of the transformed decomposition technique. The decomposition tree developed for circular-arc graphs generalizes the concept of the PQ-tree, which is a data structure that keeps track of all possible interval representations of a given interval graph. As a consequence, our approach also yields an $O(m \cdot n)$ isomorphism algorithm for circle graphs.