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 >= 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)) boolean AND O(n(3.5)) time, or O(m(2)) boolean AND O(n(3.5)) without fast matrix multiplication. (C) 2013 Elsevier B.V. All rights reserved.