Quasi-kernels and quasi-sinks

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $D=(V,A)$ be a digraph, where $V$ is any infinite set. A set is independent if no arc joins two of its vertices. In this question a quasi-kernel $K$ is an independent set such that every vertex is reachable from some member of $K$ by a finite directed path of length at most two. A quasi-sink is a quasi-kernel after all arcs are reversed. Paths of length zero are allowed. For $W\subseteq V$, $D[W]$ denotes the induced digraph.

Statement recorded in the supplied source.

Is there always a partition $V=V_1\dot\cup V_2$ such that $D[V_1]$ has a quasi-kernel and $D[V_2]$ has a quasi-sink? Either part may be empty, in which case its empty set supplies the required object. [S2]

Short English statement

Can every infinite digraph be partitioned into an induced part dominated in two steps by an independent set and a part satisfying the reversed condition?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3000067 (AMR-029-0067). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 P. L. Erdős and L. Soukup, Quasi-kernels and quasi-sinks in infinite graphs (2007), Introduction, definitions and Conjecture 1.2, pp.1--2. https://arxiv.org/pdf/0712.0663.