Lower General Position in Cartesian Products

TL;DR

This paper investigates the lower general position number in Cartesian product graphs, establishing its values for various families, and introduces the concept of terminal sets to understand the structure of minimal maximal general position sets.

Contribution

It determines the lower general position numbers for several Cartesian product families and introduces terminal sets, linking their existence to graph diameter and providing bounds.

Findings

Lower general position numbers are determined for specific graph families.

Existence of terminal sets is proven for graphs with diameter at most three.

Conjectures relate terminal sets to general position properties in all graphs.

Abstract

A subset $S$ of vertices of a graph $G$ is in \emph{general position} if no shortest path in $G$ contains three vertices of $S$. The \emph{general position problem} consists of finding the number of vertices in a largest general position set of $G$, whilst the \emph{lower general position problem} asks for a smallest maximal general position set. In this paper we determine the lower general position numbers of several families of Cartesian products. We also show that the existence of small maximal general position sets in a Cartesian product is connected to a special type of general position set in the factors, which we call a \emph{terminal set}, for which adding any vertex $u$ from outside the set creates three vertices in a line with $u$ as an endpoint. We give a constructive proof of the existence of terminal sets for graphs with diameter at most three. We also present conjectures on the existence of terminal sets for all graphs and a lower bound on the lower general position number of a Cartesian product in terms of the lower general position numbers of its factors.