We prove that every digraph of circumference l has DAG-width at most l and this is best possible. As a consequence of our result we deduce that the k-linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in [2] . We also prove that the weak k-linkage problem (where we ask for arc-disjoint paths) is polynomially solvable for every fixed k in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is open. Finally we prove that the minimum spanning strong subdigraph problem is NP-hard on digraphs of DAG-width at most 5.
Moreover, the arc version of the k-linkage problem is fixed-parameter tractable on digraphs of directed circumference 2 REF ; the question remains open for digraphs of arbitrary directed circumference.
DAG-width and circumference of digraphs
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Mathematics Computer Science