4 research outputs found

    On the Small Quasi-kernel conjecture

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    The Chv\'atal-Lov\'asz theorem from 1974 establishes in every finite digraph GG the existence of a quasi-kernel, i.e., an independent 22-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 ∣V(G)∣/2|V(G)|/2 if GG 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

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    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
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