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

Abstract: A lemma of Fouquet implies that a claw-free graph contains an induced , 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 in a graph. Our algorithm runs in time, improving upon Shrem, Stern, and Golumbic’s recent algorithm, which uses a local approach. The best known recognition algorithms for claw-free graphs run in time, or without fast matrix multiplication. [Copyright &y& Elsevier]