Generalization of edge general position problem

TL;DR

This paper introduces the edge k-general position problem in graphs, exploring its properties, solutions, and connections to other problems, with exact values computed for various network types.

Contribution

It defines the edge k-general position problem, establishes its relationship with the edge geodesic cover problem, and provides exact solutions for specific network classes.

Findings

Exact edge k-general position numbers for torus networks

Exact values for hypercubes and Benes networks

Connections established between dual min-max problems

Abstract

The edge geodesic cover problem of a graph $G$ is to find a smallest number of geodesics that cover the edge set of $G$. The edge $k$-general position problem is introduced as the problem to find a largest set $S$ of edges of $G$ such that no $k-1$ edges of $S$ lie on a common geodesic. We study this dual min-max problems and connect them to an edge geodesic partition problem. Using these connections, exact values of the edge $k$-general position number is determined for different values of $k$ and for different networks including torus networks, hypercubes, and Benes networks.