Structural Parameterizations of $k$-Planarity

2025-06-12

AI Summary

This paper investigates the parameterized complexity of *k-planarity testing*—determining whether a graph admits a drawing where each edge is crossed at most *k* times—and of computing the *local crossing number*, i.e., the minimum such *k*. For arbitrary *k* ≥ 1, it systematically analyzes how structural parameters—including treedepth, vertex cover number, feedback vertex set number, and pathwidth—affect computational hardness. The work extends the NP-completeness of 1-planarity to near-planar graphs with feedback vertex set number ≤ 3 and pathwidth ≤ 4—the first such result for bounded structural parameters. It further proves that the local crossing number is inapproximable within any constant factor even when the feedback vertex set number is at most 2. Finally, tight upper and lower bounds for *k*-planarity are established with respect to treedepth and vertex cover number, generalizing prior results (Bannister et al., 2018) restricted to *k* = 1 and substantially broadening the parameterized framework for *k*-planarity.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $2$.

Problem

• Extends $k$-planarity parameterized complexity to general cases $k ge 1$
• Proves NP-completeness for near-planar graphs with bounded pathwidth
• Shows hardness of approximating local crossing number for constant-treewidth graphs

Innovation

• Extends $k$-planarity to general case $k ge 1$
• Uses structural parameterizations like treedepth
• Provides tight parameterized bounds results