Recognizing (Unit) Interval Graphs by Zigzag Graph Searches

TL;DR

This paper simplifies the recognition algorithms for interval and unit interval graphs, providing shorter proofs, easier implementations, and revealing that fewer graph search sweeps are sufficient for recognition.

Contribution

It offers a new, shorter proof and simpler implementation for Li and Wu's interval graph recognition algorithm, and shows that only two sweeps suffice for unit interval graphs recognition.

Findings

Li and Wu's algorithm can be simplified with fewer graph search sweeps.

Two sweeps are sufficient for recognizing unit interval graphs.

New structural observations enhance understanding of graph recognition algorithms.

Abstract

Corneil, Olariu, and Stewart [SODA 1998; SIAM Journal on Discrete Mathematics 2009] presented a recognition algorithm for interval graphs by six graph searches. Li and Wu [Discrete Mathematics \& Theoretical Computer Science 2014] simplified it to only four. The great simplicity of the latter algorithm is however eclipsed by the complicated and long proofs. The main purpose of this paper is to present a new and significantly short proof for Li and Wu's algorithm, as well as a simpler implementation. We also give a self-contained simpler interpretation of the recognition algorithm of Corneil [Discrete Applied Mathematics 2004] for unit interval graphs, based on three sweeps of graph searches. Moreover, we show that two sweeps are already sufficient. Toward the proofs of the main results, we make several new structural observations that might be of independent interests.