Graph theory general position problem

TL;DR

This paper introduces a graph theory variation of the general position problem, exploring the maximum size of vertex sets with no three collinear on a geodesic, providing bounds, exact values for certain graphs, and proving NP-completeness.

Contribution

It defines the gp-number for graphs, establishes upper bounds, computes it for specific classes, and proves the problem's NP-completeness.

Findings

Derived upper bounds on gp(G) using isometric covers

Determined gp-number for several graph classes

Proved the general position problem is NP-complete

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 $S'$ of $S$ such that no three points of $S'$ 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 ${\rm gp}(G)$ of $G$. Upper bounds on ${\rm gp}(G)$ 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.