Monotone Clustered Level Planarity

2026-07-20

AI Summary

This study addresses the monotone clustered level planarity (mCLP) problem, which seeks to augment a given hierarchical graph with edges so that each cluster becomes connected while preserving level planarity and without introducing crossings across cluster boundaries. The work presents the first systematic analysis of the parameterized complexity of mCLP, highlighting the computational challenges arising from its non-hereditary nature. By combining techniques from parameterized complexity theory, graph reduction, and structural parameter analysis, the authors prove that mCLP remains NP-complete even when restricted to forests, graphs of bounded treewidth, or instances with few clusters or levels. However, they establish the first positive algorithmic result by showing that mCLP is fixed-parameter tractable when parameterized jointly by the vertex cover number and the number of clusters.

Abstract

We consider the combination of the two constrained planarity problems Level- and Clustered Planarity. Traditionally, level-planar drawings with convex clusters have been studied in this setting. Fink et al. (EuroCG 2024) recently introduced a different way of combining level- and clustered planarity by mimicking a classic characterization of clustered planarity in the level-planar setting: The problem (y-)monotone Clustered Level Planarity (mCLP) seeks a level-planar drawing in which it is possible to augment each cluster with edges that do not cross cluster boundaries so that it becomes connected while maintaining level-planarity. This is in line with previous research on clustered planarity that poses certain requirements on the augmentation edges that make each cluster connected, e.g., that they form a path. Fink et al. (EuroCG 2024) showed that mCLP is NP-complete even for biconnected single-source graphs and instances with a constant number of levels and clusters. We further classify the parameterized complexity of the mCLP problem by, on the one hand, showing hardness even for instances that consist of a forest with trees of bounded size, no isolated vertices, and a small constant number of either clusters or levels. This excludes fixed-parameter tractability for almost all graph-structural parameters, except for vertex cover, even in conjunction with the number of clusters. We complement this by showing fixed-parameter tractability when parameterizing by the vertex cover number and the number of clusters. A major obstacle is the fact that mCLP is non-hereditary, i.e., subinstances of yes-instances may be no-instances and vice versa, which makes it challenging to apply usual reduction techniques.

Problem

• monotone Clustered Level Planarity
• level planarity
• clustered planarity
• parameterized complexity
• NP-completeness

Innovation

• monotone Clustered Level Planarity
• parameterized complexity
• fixed-parameter tractability
• vertex cover
• non-hereditary