Quasi-kernels and quasi-sinks in infinite graphs

Given a directed graph G = ( V , E ) an independent set A ⊂ V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A ). Every finite directed graph has a quasi-kernel. The plain generalization for infinite graphs fails, even for tournaments. We study the following conjecture: for any digraph G = ( V , E ) there is a a partition ( V 0 , V 1 ) of the vertex set such that the induced subgraph G [ V 0 ] has a quasi-kernel and the induced subgraph G [ V 1 ] has a quasi-sink.