Fully dynamic recognition of proper circular-arc graphs

TL;DR

This paper introduces a fully dynamic algorithm for recognizing proper circular-arc graphs, supporting efficient vertex and edge updates, and providing fast algorithms for related connectivity and co-bipartiteness problems.

Contribution

It presents the first fully dynamic recognition algorithm for PCA graphs with efficient update times and auxiliary algorithms for connectivity and co-bipartiteness.

Findings

Edge operations run in O(log n) time.

Vertex operations run in O(log n + d) time.

Co-bipartiteness checks are performed in O(Δ) time.

Abstract

We present a fully dynamic algorithm for the recognition of proper circular-arc (PCA) graphs. The allowed operations on the graph involve the insertion and removal of vertices (together with its incident edges) or edges. Edge operations cost O(log n) time, where n is the number of vertices of the graph, while vertex operations cost O(log n + d) time, where d is the degree of the modified vertex. We also show incremental and decremental algorithms that work in O(1) time per inserted or removed edge. As part of our algorithm, fully dynamic connectivity and co-connectivity algorithms that work in O(log n) time per operation are obtained. Also, an O(\Delta) time algorithm for determining if a PCA representation corresponds to a co-bipartite graph is provided, where \Delta\ is the maximum among the degrees of the vertices. When the graph is co-bipartite, a co-bipartition of each of its co-components is obtained within the same amount of time.