Unbent collections of non-planar $s$-grid-drawing

2026-06-25

AI Summary

This study addresses the problem of covering all edges of a given graph with the minimum number of non-planar orthogonal drawings such that every edge appears without bends in at least one drawing. Extending the notion of bend-free edge sets from planar to non-planar settings, the work focuses on orthogonal drawings over $s$-grids. The main contribution is the first proof that any graph admits a bend-cover using only two $s$-grid orthogonal drawings for $s \geq 3$, improving upon the known lower bound of three required in the planar case. This result is established through a combination of graph-theoretic arguments and orthogonal graph drawing techniques, and its generality is demonstrated across multiple graph classes, thereby establishing a tight upper bound of two for the bend-cover number in non-planar $s$-grid orthogonal representations.

Abstract

In a recent paper, Antić et al.~studied collections of planar orthogonal drawings of a graph where every edge is unbent in at least one drawing. This paper generalizes this concept to non-planar drawings, and shows that then two drawings always suffice (for planar drawings three drawings are sometimes needed). The results can also be generalized to $s$-grid drawings for $s\geq 3$.

Problem

• non-planar
• orthogonal drawing
• unbent edge
• s-grid drawing
• graph representation

Innovation

• non-planar drawing
• unbent edge
• s-grid drawing
• orthogonal graph drawing
• graph representation