On the General Position Number of Mycielskian Graphs

Abstract

The general position problem for graphs was inspired by the no-three-in-line problem and the general position subset selection problem from discrete geometry. A set S of vertices of a graph G is a general position set if no shortest path between two vertices of S contains a third element of S; the general position number of G is the size of a largest general position set. In this note we investigate the general position numbers of the Mycielskian of graphs, which are of interest in chromatic graph theory. 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.