A Faster Algorithm to Recognize Even-Hole-Free Graphs

Abstract: We study the problem of determining whether an $n$-node graph $G$ has an even hole, i.e., an induced simple cycle consisting of an even number of nodes. Conforti, Cornu\'ejols, Kapoor, and Vu\v{s}kovi\'c gave the first polynomial-time algorithm for the problem, which runs in $O(n<sup>{40})$ time. Later, Chudnovsky, Kawarabayashi, and Seymour reduced the running time to $O(n<sup>{31})$. The best previously known algorithm for the problem, due to da Silva and Vu\v{s}kovi\'c, runs in $O(n<sup>{19})$ time. In this paper, we solve the problem in $O(n<sup>{11})$ time. Moreover, if $G$ has even holes, our algorithm also outputs an even hole of $G$ in $O(n<sup>{11})$ time.