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.

In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex x of a connected graph G, we say that a set S ⊆ V (G) is an x-position set if for any y ∈ S the shortest x, y-paths in G contain no point of S \ {y}. We investigate the largest and smallest orders of maximum x-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.