Drawing graphs with vertices and edges in convex position

TL;DR

This paper investigates graphs with vertices and edges in convex position, proving an upper edge bound of 2n-3, presenting non-planar examples, and exploring implications for convex sets in Minkowski sums.

Contribution

It establishes a tighter upper bound on edges for graphs of strong convex dimension 2 and introduces non-planar examples, advancing understanding of their properties.

Findings

Graphs of strong convex dimension 2 have at most 2n-3 edges.

Existence of non-planar graphs with strong convex dimension 2.

Results inform bounds on convexly independent sets in Minkowski sums.

Abstract

A graph has strong convex dimension $2$, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension $2$ are planar and therefore have at most $3n-6$ edges. We prove that all such graphs have at most $2n-3$ edges while on the other hand we present a class of non-planar graphs of strong convex dimension $2$. We also give lower bounds on the maximum number of edges a graph of strong convex dimension $2$ can have and discuss variants of this graph class. We apply our results to questions about large convexly independent sets in Minkowski sums of planar point sets, that have been of interest in recent years.