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

W. Sean Kennedy, Andrew D. King
2011 arXiv   pre-print
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
more » ... hm runs in O(m^2+n^2 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) ∩ O(n^3.5) time, or O(m^2) ∩ O(n^3.5) without fast matrix multiplication.
arXiv:1103.6222v2 fatcat:y4oa7m5f2fhuhkqqyknoiaumsi