Lower General Position in Cartesian Products

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Kruft Welton Eartha; Khudairi Sharif; Tuite James

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Journal Paper

Paper Information

Journal: COMMUNICATIONS IN COMBINATORICS AND OPTIMIZATION Year:2025 | Volume:10 | Issue:1 Page(s): 110-125

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Information Journal Paper

Title

Lower General Position in Cartesian Products

Author(s)

Kruft Welton Eartha | Khudairi Sharif | Tuite James | Issue Writer Certificate

Keywords

general position number

universal line

Cartesian product

Abstract

A subset $S$ of vertices of a graph $G$ is in general position if no shortest path in $G$ contains three vertices of $S$. The general position problem consists of finding the number of vertices in a largest general position set of $G$, whilst the 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 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.

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