Moving through Cartesian products, coronas and joins in general position

TL;DR

This paper studies the mobile general position problem, where robots visit all vertices in a graph while maintaining general position, focusing on Cartesian products, coronas, and joins, providing bounds and exact values for various graph families.

Contribution

It introduces bounds and exact solutions for the mobile general position problem on complex graph constructions like Cartesian products, coronas, and joins.

Findings

Exact values for grids, cylinders, Hamming graphs, and prisms of trees.

Upper and lower bounds for general graphs.

Insights into the mobile general position problem in complex graph structures.

Abstract

The general position problem asks for large sets of vertices such that no three vertices of the set lie on a common shortest path. Recently a dynamic version of this problem was defined, called the \emph{mobile general position problem}, in which a collection of robots must visit all the vertices of the graph whilst remaining in general position. In this paper we investigate this problem in the context of Cartesian products, corona products and joins, giving upper and lower bounds for general graphs and exact values for families including grids, cylinders, Hamming graphs and prisms of trees.