Lai, K.-Y.K.-Y.LaiThorup, M.M.ThorupHSUEH-I LU2021-05-052021-05-052020https://www.scopus.com/inward/record.url?eid=2-s2.0-85086758711&partnerID=40&md5=b597968accc9b091fb419c1b0d8ebb13https://scholars.lib.ntu.edu.tw/handle/123456789/558949The three-in-a-tree problem is to determine if a simple undirected graph contains an induced subgraph which is a tree connecting three given vertices. Based on a beautiful characterization that is proved in more than twenty pages, Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known polynomial-time algorithm, running in O(mn2) time, to solve the three-in-a-tree problem on an n-vertex m-edge graph. Their three-in-a-tree algorithm has become a critical subroutine in several state-of-the-art graph recognition and detection algorithms. In this paper we solve the three-in-a-tree problem in O(mlog2 n) time, leading to improved algorithms for recognizing perfect graphs and detecting thetas, pyramids, beetles, and odd and even holes. Our result is based on a new and more constructive characterization than that of Chudnovsky and Seymour. Our new characterization is stronger than the original, and our proof implies a new simpler proof for the original characterization. The improved characterization gains the first factor n in speed. The remaining improvement is based on dynamic graph algorithms. © 2020 ACM.Dynamic graph algorithm; Even hole; Graph recognition; Induced subgraph detection; Odd hole; Perfect graph; SPQR-tree; Top treeconference paper10.1145/3357713.33842352-s2.0-85086758711