A note on quasi-kernels in digraphs

Abstract

We describe a simple linear time algorithm to construct a quasi-kernel in a digraph and to find three quasi-kernels in digraphs without a kernel (giving constructive proofs of known results of Chvátal and Lovász, or Jacob and Meyniel). However, we show that it is NP-complete to decide if there is a quasi-kernel containing a specified vertex in a given digraph. The algorithm provides also a simple proof of the characterization of digraphs with only two quasi-kernels given by Gutin, Koh, Tay and Yeo.