The 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 \(\mathrm{gp}\)-set of G, and its cardinality is the \(\mathrm{gp}\)-number, \(\mathrm{gp}(G)\), 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, \(\mathrm{gp_{e}}(G)\), is the size of a largest edge general position set of G. For r-dimensional hypercube \(Q_r\), it is proved that \(\mathrm{gp_{e}}(Q_r) = 2^r\), and for arbitrary tree T, it is shown that \(\mathrm{gp_{e}}(T)\) is the number of its leaves. The value of \(\mathrm{gp_{e}}(P_r\, \square \, P_s)\) 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.