A complete characterization of split digraphs with a strong arc decomposition
Abstract
A strong arc decomposition of a (multi-)digraph D(V, A) is a partition of its arc set A into two disjoint arc sets A1 and A2 such that both of the spanning subdigraphs D(V, A1) and D(V, A2) are strong. In this paper, we fully characterize all split digraphs that do not have a strong decomposition. This resolves two problems proposed by Bang-Jensen and Wang and contributes to a series of efforts aimed at addressing this problem for specific graph classes. This work continues the research on semicomplete composition [Bang-Jensen, Gutin and Yeo, J. Graph Theory, 2020]; on locally semicomplete digraphs [Bang-Jensen and Huang, J. Combin. Theory Ser. B, 2010]; on a type of tournaments [Bang-Jensen and Yeo, Combinatorica, 2004].
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