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 $C_5$, 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 $\geq 5$ in a graph. Our algorithm runs in $O(m^2+n^2\log n)$ time, improving upon Shrem, Stern, and Golumbic's recent $O(nm^2)$ algorithm, which uses a local approach. The best known recognition algorithms for claw-free graphs run in $O(m^{1.69}) \cap O(n^{3.5})$ time, or $O(m^2) \cap O(n^{3.5})$ without fast matrix multiplication.