4 research outputs found
On the Small Quasi-kernel conjecture
The Chv\'atal-Lov\'asz theorem from 1974 establishes in every finite digraph
the existence of a quasi-kernel, i.e., an independent -out-dominating
vertex set. In the same spirit, the Small Quasi-kernel Conjecture, proposed by
Erd\H{o}s and Sz\'ekely in 1976, asserts the existence of a quasi-kernel of
order at most if does not have sources. Despite repeated
efforts, the conjecture remains wide open.
This work contains a number of new results towards the conjecture. In our
main contribution we resolve the conjecture for all directed graphs without
sources containing a kernel in the second out-neighborhood of a quasi-kernel.
Furthermore, we provide a novel strongly connected example demonstrating the
asymptotic sharpness of the conjecture. Additionally, we resolve the conjecture
in a strong form for all directed unicyclic graphs.Comment: 12 pages, 1 figur
Quasi-kernels and quasi-sinks in infinite graphs
AbstractGiven a directed graph G=(V,E) an independent set A⊂V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed graph has a quasi-kernel. The plain generalization for infinite graphs fails, even for tournaments. We study the following conjecture: for any digraph G=(V,E) there is a a partition (V0,V1) of the vertex set such that the induced subgraph G[V0] has a quasi-kernel and the induced subgraph G[V1] has a quasi-sink