Quasi-kernels and quasi-sinks in infinite graphs

TL;DR

This paper explores the existence of quasi-kernels and quasi-sinks in infinite directed graphs, proposing a conjecture about partitioning vertices to ensure each part contains one, extending finite graph properties.

Contribution

It introduces a conjecture that every infinite directed graph can be partitioned into parts with quasi-kernels and quasi-sinks, generalizing finite graph results.

Findings

Finite graphs always have a quasi-kernel.

The generalization to infinite graphs fails for some cases.

The paper investigates the conjecture for infinite graphs.

Abstract

Given a directed graph G=(V,E) an independent set A of the vertices 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 investigate the following conjecture here: for any digraph G=(V,E) there is a a partition (V_0,V_1) of the vertex set such that the induced subgraph G[V_0] has a quasi-kernel and the induced subgraph G[V_1] has a quasi-sink.