A note on quasi-kernels in digraphs

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. A linear time algorithm to construct a quasi-kernel (q-k) in a digraph is presented.The algorithm is used to find three q-ks in digraphs without a kernel.Also it provides a short proof of the structure of digraphs with 2 q-ks.We proved that deciding if there is a q-k containing a given vertex is NP-complete.