For an integer $t \geq 1$, a homomorphism of a digraph G to a digraph $H$ is $t$-frugal if no more than $t$ in-neighbours of any vertex of $G$ have the same image. There is a dichotomy theorem based on structural properties when $t=1$ and $H$ has a loop at each vertex, but there is unlikely to be such a theorem for general digraphs $H$. For $t \geq 2$ we describe a structural dichotomy theorem for $t$-frugal homomorphisms of general digraphs.

翻译：对于整数 $t \geq 1$，有向图 $G$ 到有向图 $H$ 的同态称为 $t$-节俭的，如果 $G$ 中任意顶点的入邻域中，具有相同像的顶点数不超过 $t$ 个。当 $t=1$ 且 $H$ 在每个顶点处都有自环时，存在一个基于结构性质的二分法定理，但对于一般有向图 $H$，此类定理不太可能存在。对于 $t \geq 2$，我们描述了一个关于一般有向图 $t$-节俭同态的结构二分法定理。