Structural Parameterizations of $k$-Planarity

TL;DR

This paper explores the computational complexity of $k$-planarity in graphs, providing new parameterized bounds and hardness results for various graph classes, extending previous work focused on the case $k=1$.

Contribution

It generalizes the parameterized complexity analysis of $k$-planarity to all $k \, \ge \, 1$, establishing tight bounds and hardness results for subclasses of graphs.

Findings

Testing 1-planarity is NP-complete for near-planar graphs with small feedback vertex set.

Local crossing number approximation is hard within any constant factor for graphs with small feedback vertex set.

Provides tight bounds on the complexity of $k$-planarity for graphs with bounded treewidth and related parameters.

Abstract

The concept of $k$-planarity is extensively studied in the context of Beyond Planarity. A graph is $k$-planar if it admits a drawing in the plane in which each edge is crossed at most $k$ times. The local crossing number of a graph is the minimum integer $k$ such that it is $k$-planar. The problem of determining whether an input graph is $1$-planar is known to be NP-complete even for near-planar graphs [Cabello and Mohar, SIAM J. Comput. 2013], that is, the graphs obtained from planar graphs by adding a single edge. Moreover, the local crossing number is hard to approximate within a factor $2 - \varepsilon$ for any $\varepsilon > 0$ [Urschel and Wellens, IPL 2021]. To address this computational intractability, Bannister, Cabello, and Eppstein [JGAA 2018] investigated the parameterized complexity of the case of $k = 1$, particularly focusing on structural parameterizations on input graphs, such as treedepth, vertex cover number, and feedback edge number. In this paper, we extend their approach by considering the general case $k \ge 1$ and give (tight) parameterized upper and lower bound results. In particular, we strengthen the aforementioned lower bound results to subclasses of constant-treewidth graphs: we show that testing $1$-planarity is NP-complete even for near-planar graphs with feedback vertex set number at most $3$ and pathwidth at most $4$, and the local crossing number is hard to approximate within any constant factor for graphs with feedback vertex set number at most $2$.