Min-1-Planarity is NP-Hard

TL;DR

This paper proves that deciding if a graph has a min-1-planar drawing is NP-hard, extending the understanding of graph drawing complexity for a generalized planarity concept.

Contribution

It establishes the NP-hardness of min-1-planarity testing, a previously unexplored problem in the context of generalized graph planarity.

Findings

Determined min-1-planarity decision problem is NP-hard.

Extended the class of graph planarity problems known to be computationally difficult.

Abstract

In this paper, we show that it is NP-hard to determine whether a given graph admits a min-1-planar drawing. A drawing of a graph is min-$k$-planar if, for every crossing in the drawing, at least one of the two crossing edges involves at most $k$ crossings. This notion of min-$k$-planarity was introduced by Binucci, B\"{u}ngener, Di Battista, Didimo, Dujmovi\'c, Hong, Kaufmann, Liotta, Morin, and Tappini [GD 2023; JGAA, 2024] as a generalization of $k$-planarity.