On the General Position Number of Mycielskian Graphs

TL;DR

This paper studies the maximum size of vertex sets in Mycielskian graphs where no three vertices lie on a shortest path, providing bounds and exact values for various graph classes.

Contribution

It establishes tight bounds on the general position number of Mycielskian graphs and determines exact values for specific graph classes.

Findings

Tight bounds on the general position number for Mycielskian graphs.

Exact values computed for cubic graphs and trees.

Structural insights into graphs meeting these bounds.

Abstract

The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of $S$. The \emph{general position number} of $G$ is the number of vertices in a largest general position set. In this paper we investigate the general position numbers of the Mycielskian of graphs. We give tight upper and lower bounds on the general position number of the Mycielskian of a graph $G$ and investigate the structure of the graphs meeting these bounds. We determine this number exactly for common classes of graphs, including cubic graphs and a wide range of trees.