# The general position problem and strong resolving graph

## TL;DR

This paper investigates the general position number of graphs, establishes bounds using the strong resolving graph, and explores its behavior under strong graph products with various graph classes.

## Contribution

It proves a lower bound for the general position number using the strong resolving graph and analyzes its sharpness across multiple graph families, including strong products.

## Key findings

- The general position number is at least the clique number of the strong resolving graph.
- The bound is sharp for various graph constructions including direct and strong products.
- For strong products, the general position number satisfies a multiplicative inequality in certain cases.

## Abstract

The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three pairwise distinct vertices from $S$ lie on a common geodesic. It is proved that ${\rm gp}(G)\ge \omega(G_{\rm SR}$, where $G_{\rm SR}$ is the strong resolving graph of $G$, and $\omega(G_{\rm SR})$ is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs. For the strong product it is proved that $gp(G\boxtimes H) \ge gp(G)gp(H)$, and asked whether the equality holds for arbitrary connected graphs $G$ and $H$. It is proved that the answer is in particular positive for strong products with a complete factor, for strong products of complete bipartite graphs, and for certain strong cylinders.