Geodetic convexity and Kneser graphs

TL;DR

This paper studies geodetic convexity parameters in Kneser graphs, providing exact values and bounds for the geodetic number and hull number, especially for diameter two graphs, advancing understanding of their convexity properties.

Contribution

It determines the geodetic and hull numbers for Kneser graphs of diameter two, identifying exceptions and characterizing endpoints of diametral paths.

Findings

Geodetic hull number is two for most diameter two Kneser graphs.

Exceptions include K(5,2), K(6,2), and K(8,2) with hull number three.

Characterization of diametral path endpoints aids in deriving main results.

Abstract

The {\em Kneser graph} $K(2n+k,n)$, for positive integers $n$ and $k$, is the graph $G=(V,E)$ such that $V=\{S\subseteq\{1,\ldots,2n+k\} : |S|=n\}$ and there is an edge $uv\in E$ whenever $u\cap v=\emptyset$. Kneser graphs have a nice combinatorial structure, and many parameters have been determined for them, such as the diameter, the chromatic number, the independence number, and, recently, the hull number (in the context of $P_3$-convexity). However, the determination of geodetic convexity parameters in Kneser graphs still remained open. In this work, we investigate both the geodetic number and the geodetic hull number of Kneser graphs. We give upper bounds and determine the exact value of these parameters for Kneser graphs of diameter two (which form a nontrivial subfamily). We prove that the geodetic hull number of a Kneser graph of diameter two is two, except for $K(5,2)$, $K(6,2)$, and $K(8,2)$, which have geodetic hull number three. We also contribute to the knowledge on Kneser graphs by presenting a characterization of endpoints of diametral paths in $K(2n+k,n)$, used as a tool for obtaining some of the main results in this work.