6 problems

Let $D$ be a digraph. For an integer $s\ge 1$, call a set $S\subseteq V(D)$ an $s$-source set if … where $N^-(S)$ is the set of external in-neighbors of $S$. For an integer…

Let $D$ be a source-free bipartite digraph, meaning that every vertex has a nonempty set of external in-neighbors. A quasikernel of $D$ is an independent set $Q\subseteq V(D)$ such…

Large-quasikernel structural conjecture. Every digraph $D$ either contains a subset $A\subseteq V(D)$ such that $D[A]$ is acyclic and

Minimum in-degree quasikernel conjecture. There exists an integer $\delta\ge 2$ such that every digraph $D$ with $\delta^-(D)\ge\delta$ contains a quasikernel $Q$ with

Large Quasikernel Conjecture. Every digraph $D$ contains a quasikernel $Q$ with

Small Quasikernel Conjecture. If $D$ is a source-free digraph, then $D$ contains a quasikernel $Q$ with