Finding a smallest odd hole in a claw-free graph using global structure

A lemma of Fouquet implies that a claw-free graph contains an induced C5C5, contains no odd hole, or is quasi-line. In this paper, we use this result to give an improved shortest-odd-hole algorithm for claw-free graphs by exploiting the structural relationship between line graphs and quasi-line graphs suggested by Chudnovsky and Seymour’s structure theorem for quasi-line graphs. Our approach involves reducing the problem to that of finding a shortest odd cycle of length ≥5≥5 in a graph. Our algorithm runs in O(m2+n2logn)O(m2+n2logn) time, improving upon Shrem, Stern, and Golumbic’s recent O(nm2)O(nm2) algorithm, which uses a local approach. The best known recognition algorithms for claw-free graphs run in O(m1.69)∩O(n3.5)O(m1.69)∩O(n3.5) time, or O(m2)∩O(n3.5)O(m2)∩O(n3.5) without fast matrix multiplication.