Kelly-width is a parameter of digraphs recently proposed by Hunter and Kreutzer as a directed analogue of treewidth. We give an alternative characterization of digraphs of bounded Kelly-width in support of this analogy, and the first polynomial-time algorithm recognizing digraphs of Kelly-width 2. For an input digraph G = (V, A) the algorithm outputs a vertex ordering and a digraph H = (V, B) with A subset of B witnessing either that G has Kelly-width at most 2 or that G has Kelly-width at least 3, in time linear in H.
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