Seminar

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Given a digraph $D$, we say that a set of vertices $Q\subseteq V(D)$ is a quasi-kernel if $Q$ is an independent set and if every vertex of $D$ can be reached from $Q$ by a path of length at most 2.  The Small Quasi-kernel Conjecture of P.L.\ Erd\H{o}s and Sz\'ekely from 1976 states that every $n$-vertex source-free digraph $D$ contains a quasi-kernel of size at most $\frac{1}{2}n$.  Despite being posed nearly 50 years ago, very little is known about this conjecture, with the only non-trivial upper bound of $n-\frac{1}{4}\sqrt{n\log n}$ being proven recently by ourself.  We discuss this result together with a number of other related results and open problems around the Small Quasi-kernel Conjecture.