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For nonnegative integers $k$ and $l$, let $\mathscr{D}(k,l)$ denote thefamily of digraphs in which every vertex has either indegree at most $k$ oroutdegree at most $l$. In this paper we prove that the edges of every digraphin $\mathscr{D}(3,3)$ and $\mathscr{D}(4,4)$ can be covered by at most fivedirected cuts and present an example in $\mathscr{D}(3,3)$ showing that thisresult is best possible.

Covering the edges of digraphs in $\mathscr{D}(3,3)$ and $\mathscr{D}(4,4)$ with directed cuts