# Quasi-kernels and quasi-sinks

A quasi-kernel of a digraph $D$ is an independent vertex set $K$ sucht that every vertex is reachable from $K$ in $D$ by a path of length at most two. A quasi-sink of $D$ is a quasi-kernel of the digraph that we obtain by changing the direction of the edges in $D$. Is it true that for any infinite digraph $D=(V,A)$ there is a partition $\{V_1, V_2\}$ of $V$ such that $D[V_1]$ admits a quasi-kernel and $D[V_2]$ admits a quasi-sink?
V. Chvátal and L. Lovász proved that every finite digraph has a quasi-kernel [1]. This is not true for infinite digraphs, take for example $V=\mathbb{Z}$ and let $(m,n)\in A$ iff $m\ltn$. P. L. Erdős, A. Hajnal, L. Soukup showed that in any infinite digraph, one can find disjoint independent sets $S, T$ such that for every vertex $v$ there is a path $P$ of length at most two such that $P$ either goes from $S$ to $v$ or from $v$ to $T$ ([2] p. 3041. Theorem 2.1).