On some extremal position problems for graphs

The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers, with applications to a realisation result. We then solve a Turan problem for the size of graphs with given order and position numbers and characterise the possible diameters of graphs with given order and monophonic position number. Finally we classify the graphs with given order and diameter and largest possible general position number.