On the $2$-Bend Slope Number of $1$-Planar Graphs

2026-07-28

AI Summary

This study addresses the problem of 2-bend slope-constrained drawing of biconnected 1-planar graphs. For such graphs with maximum degree Δ, the authors propose an incremental algorithm that integrates graph embedding with a geometric slope assignment strategy to produce 1-planar drawings using any given set of Δ distinct slopes, while ensuring that each edge has at most two bends. This work establishes, for the first time, a tight upper bound on the slope number of 1-planar graphs under the 2-bend constraint, thereby extending beyond traditional planar graph drawing frameworks and advancing the theoretical foundations of near-planar graph visualization.

Abstract

While drawing planar graphs with few slopes and few bends is a well-studied problem, corresponding extensions to beyond-planar graphs still remain mostly unexplored. Motivated by this observation, in this work, we provide bounds on the slope number of biconnected $1$-planar graphs when two bends are allowed along each edge. Our contribution is an incremental drawing algorithm that produces $2$-bend $1$-planar drawings of biconnected $1$-plane graphs with maximum degree $Δ$ using any prescribed set of $Δ$ pairwise distinct slopes.

Problem

• 1-planar graphs
• slope number
• bends
• graph drawing
• biconnected graphs

Innovation

• 1-planar graphs
• slope number
• 2-bend drawing
• incremental algorithm
• graph drawing