Weakly Leveled Planarity with Bounded Span

2024-09-03

International Symposium Graph Drawing and Network Visualization

AI Summary

This paper investigates the *s-span weakly layered planar embedding* problem: vertices must lie on horizontal layers, and edges must be *y*-monotone curves spanning at most *s* layers. We establish the first comprehensive parameterized complexity landscape: the problem is para-NP-hard; we provide a polynomial kernel parameterized by vertex cover number and an FPT algorithm parameterized by tree depth. For cycle trees—graphs formed by adding a chord to a cycle—we determine the tight asymptotic span bound Θ(log *n*) and prove that all 3-connected cycle trees admit 4-span weakly layered planar embeddings—the first nontrivial constant-span result for a natural graph class. Additionally, we improve upper bounds on edge-length ratios for related graph classes. Our approach integrates techniques from parameterized algorithms, structural graph parameter analysis, combinatorial construction, and planar embedding theory.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly $y$-monotone curve. A graph is $s$-span weakly leveled planar if it admits such a drawing where the edges have span at most $s$; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing $s$-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter $s$ and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of $2$-outerplanar graphs generalizing Halin graphs, are $Theta(log n)$-span weakly leveled planar and $4$-span weakly leveled planar when $3$-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration.

Problem

• Planar Graphs
• Hierarchical Clustering
• Spanning Tree

Innovation

• s-span weak level planar graphs
• log n upper bound
• 3-connected graphs