Abstract
We prove that the minimum weight of a dicycle is equal to the maximum number of disjoint dicycle covers, for every weighted digraph whose underlying graph is planar and does not have K 5–e as a minor (K 5–e is the complete graph on five vertices, minus one edge). Equality was previously known when forbidding K 4 as a minor, while an infinite number of weighted digraphs show that planarity does not guarantee equality. The result also improves upon results known for Woodall’s Conjecture and the Edmonds-Giles Conjecture for packing dijoins. Our proof uses Wagner’s characterization of planar 3-connected graphs that do not have K 5–e as a minor.
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Lee, O., Williams, A. (2006). Packing Dicycle Covers in Planar Graphs with No K 5–e Minor. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_62
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DOI: https://doi.org/10.1007/11682462_62
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