On March 17, 2021, at the Virtual Discrete Math Colloquium, Yixin Cao (操宜新) from Hong Kong Polytechnic University gave an online talk on the recognition algorithm for interval graphs by using 4 graph searchers with an improved presentation compared to the version of Li and Wu. The title of his talk was “Recognizing (unit) interval graphs by zigzag graph searches“.
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 shorter proof for Li and Wu’s algorithm, as well as a simpler implementation. We also give a self-contained presentation 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.