On RAC Drawings of Graphs with Two Bends per Edge

TL;DR

This paper proves a tighter upper bound on the number of edges in graphs with 2-bend RAC drawings, significantly improving previous bounds and advancing understanding of geometric graph representations.

Contribution

It establishes a new upper bound of 20n-24 edges for graphs with 2-bend RAC drawings, improving the longstanding bound of 74.2n.

Findings

Maximum edges in such graphs is at most 20n-24.

Introduces bounds on plane multigraphs with specific segment orientations.

First improvement in over 12 years for this problem.

Abstract

It is shown that every $n$-vertex graph that admits a 2-bend RAC drawing in the plane, where the edges are polylines with two bends per edge and any pair of edges can only cross at a right angle, has at most $20n-24$ edges for $n\geq 3$. This improves upon the previous upper bound of $74.2n$; this is the first improvement in more than 12 years. A crucial ingredient of the proof is an upper bound on the size of plane multigraphs with polyline edges in which the first and last segments are either parallel or orthogonal.