Quasi-kernels and quasi-sinks in infinite graphs

Given a directed graph G=(V,E)G=(V,E) an independent set A⊂VA⊂V is called quasi-kernel (quasi-sink)   iff for each point vv there is a path of length at most 2 from some point of AA to vv (from vv to some point of AA). 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)G=(V,E) there is a a partition (V0,V1)(V0,V1) of the vertex set such that the induced subgraph G[V0]G[V0] has a quasi-kernel and the induced subgraph G[V1]G[V1] has a quasi-sink.