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 ≥5 in a graph. Our algorithm runs in O(m ~2+n~2logn) 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))∩O(n~(3.5)) time, or Om~2)∩O(n ~(3.5)) without fast matrix multiplication.
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