Edge general position problem

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.