Edge general position problem

TL;DR

This paper introduces and studies the edge general position problem, determining the maximum size of edge sets with no three edges lying on a common shortest path, with exact results for hypercubes, trees, and grid graphs.

Contribution

It defines the edge general position number and provides exact values for specific graph classes, expanding the understanding of general position concepts in graph theory.

Findings

gp_e(Q_r) = 2^r for hypercubes

gp_e(T) equals the number of leaves in a tree

gp_e(P_r □ P_s) is determined for all r,s ≥ 2

Abstract

Given a graph $G$, the general position problem is to find a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is called a ${\rm gp}$-$set$ of $G$ and its cardinality is the ${\rm gp}$-$number$, ${\rm gp}(G)$, of $G$. In this paper, the edge general position problem is introduced as the edge analogue of the general position problem. The edge general position number, ${\rm gp_{e}}(G)$, is the size of a largest edge general position set of $G$. It is proved that ${\rm gp_{e}}(Q_r) = 2^r$ and that if $T$ is a tree, then ${\rm gp_{e}}(T)$ is the number of its leaves. The value of ${\rm gp_{e}}(P_r\, \square\, P_s)$ is determined for every $r,s\ge 2$. To derive these results, the theory of partial cubes is used. Mulder's meta-conjecture on median graphs is also discussed along the way.