On 2-Layer k-Matching-Planar Graphs

2026-07-22

AI Summary

This study investigates the structural properties and computational complexity of recognizing k-matching planarity in bipartite graphs under two-layer layouts. Employing graph-theoretic analysis, complexity theory, and parameterized algorithms, it establishes the first tight bounds on the pathwidth of two-layer k-matching planar graphs: an upper bound of 2k+1 and a matching lower bound via explicit constructions achieving pathwidth 3⌊k/2⌋+1. Algorithmically, the recognition problem is shown to be NP-hard even when the vertex ordering on one side is fixed, yet fixed-parameter tractable for constant k. In contrast, the variant with both sides unordered admits no constant-factor approximation unless P=NP. These results comprehensively delineate the structural limits and computational boundaries inherent to this class of graphs.

Abstract

A graph is $k$-matching-planar if it admits a drawing in the plane such that, for every edge $e$, the edges crossing $e$ contain no matching of size greater than $k$. The class of $k$-matching-planar graphs generalizes other beyond-planar graph classes, such as $k$-planar and fan-planar graphs. In a $2$-layer drawing of a bipartite graph, the vertices of the two bipartition classes are placed on two parallel horizontal lines and edges are drawn as straight-line segments between them. We prove that every graph with a $2$-layer $k$-matching-planar drawing has pathwidth at most $2k+1$. Moreover, for every $k \geq 0$, we construct a graph with a $2$-layer $k$-matching-planar drawing whose pathwidth is $3\lfloor k/2\rfloor + 1$. On the algorithmic side, we consider the one-sided recognition problem where a fixed embedding of the vertices on one side is given. We show that this problem is NP-hard. On the other hand, we prove that the problem is fixed-parameter tractable with respect to $k$. Finally, we prove that the two-sided variant cannot be approximated within any constant factor in polynomial time unless $\operatorname{P}=\operatorname{NP}$.

Problem

• k-matching-planar
• 2-layer drawing
• recognition problem
• NP-hard
• fixed-parameter tractable

Innovation

• k-matching-planar
• pathwidth
• fixed-parameter tractable
• NP-hard
• 2-layer drawing