A certifying and dynamic algorithm for the recognition of proper circular-arc graphs

Abstract

We present a dynamic algorithm for the recognition of proper circular-arc (PCA) graphs, that supports the insertion and removal of vertices (together with its incident edges). The main feature of the algorithm is that it outputs a minimally non-PCA induced subgraph when the insertion of a vertex fails. Each operation cost O(log⁡n+d) time, where n is the number vertices and d is the degree of the modified vertex. When removals are disallowed, each insertion is processed in O(d) time. The algorithm also provides two constant-time operations to query if the dynamic graph is proper Helly (PHCA) or proper interval (PIG). When the dynamic graph is not PHCA (resp. PIG), a minimally non-PHCA (resp. non-PIG) induced subgraph is obtained.