Graph theory general position problem

Abstract

The classical no-three-in-line problem is to find the maximum number of points that can be placed in the $n\times n$ grid so that no three points lie on a line. Given a set $S$ of points in an Euclidean plane, the General Position Subset Selection Problem is to find a maximum subset $S^{\prime }$ of $S$ such that no three points of $S^{\prime }$ are collinear. Motivated by these problems, the following graph theory variation is introduced: Given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a gp-set of $G$ and its size is the gp-number $\mathrm{gp}(G)$ of $G$. Upper bounds on $\mathrm{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.