Abstract
An undirected graph is called perfectly orderable if the set of its vertices admits a linear order < such that no chordless path with vertices a, b, c, d and edges ab, bc, cd has a<b and d<c. We characterize claw-free perfectly orderable graphs by nine infinite families of forbidden induced subgraphs. This result generalizes the characterization of totally balanced matrices found by Anstee and Farber, Edmonds and Lubiw, and Hoffman, Kolen and Sakarovitch. Implicit in our argument is a polynomial-time algorithm that, given any claw-free graph G, either constructs a perfect order in G or finds one of the forbidden induced subgraphs in G.
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