The full automorphism groups of general position graphs

TL;DR

This paper characterizes the full automorphism groups of general position graphs for sets with flags of size two, solving an open problem and advancing understanding of their symmetry properties.

Contribution

It provides a complete characterization of automorphism groups of these graphs when the flag type has size two, addressing an open question in the field.

Findings

Automorphism groups are fully characterized for |T|=2.

The paper solves an open problem proposed by Klaus Metsch.

Results enhance understanding of symmetry in general position graphs.

Abstract

Let $S$ be a non-empty finite set. A flag of $S$ is a set $f$ of non-empty proper subsets of $S$ such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. The set $\{|X|:X\in f\}$ is called the type of $f$. Two flags $f$ and $f'$ are in general position with respect to $S$ if $X\cap Y=\emptyset$ or $X\cup Y=S$ for all $X\in f$ and $Y\in f'$. For a fixed type $T$, Klaus Metsch defined the general position graph $\Gamma(S,T)$ whose vertices are the flags of $S$ of type $T$ with two vertices being adjacent when the corresponding flags are in general position. In this paper, we characterize the full automorphism groups of $\Gamma(S,T)$ in the case that $|T|=2$. In particular, we solve an open problem proposed by Klaus Metsch.