Critical Kernel Imperfectness in $4$-quasi-transitive and $4$-anti-transitive digraphs of small diameter

TL;DR

This paper characterizes certain classes of asymmetric digraphs with bounded diameter that are critical kernel imperfect, meaning they lack a kernel but all their proper induced subdigraphs have one.

Contribution

It provides a characterization of asymmetrical 4-quasi-transitive, 4-transitive, 2-anti-transitive, and 4-anti-transitive digraphs with bounded diameter that are critical kernel imperfect.

Findings

Identifies classes of critical kernel imperfect digraphs within specified transitivity conditions.

Provides structural characterizations for these classes.

Enhances understanding of kernel properties in complex digraphs.

Abstract

A kernel in a digraph is an independent and absorbent subset of its vertex set. A digraph is critical kernel imperfect if it does not have a kernel, but every proper induced subdigraph does. In this article, we characterize asymmetrical $4$-quasi-transitive and $4$-transitive digraphs, as well as $2$-anti-transitive, and asymmetrical $4$-anti-transitive digraphs with bounded diameter, which are critical kernel imperfect.