The Price of Upwardness

2024-09-02

International Symposium Graph Drawing and Network Visualization

AI Summary

This paper investigates the upward $k$-planarity problem for directed acyclic graphs (DAGs): drawing a DAG such that all edges are $y$-monotone (i.e., upward) and each edge is crossed at most $k$ times ($k geq 1$). Focusing on subclasses—including bipartite outerplanar, cubic, and bounded pathwidth DAGs—the work systematically characterizes the existence thresholds and computational complexity of upward $k$-planarity. Methodologically, it employs combinatorial graph analysis, embedding theory, and algorithmic design. Key contributions include: (i) the first proof that upward $1$-planarity testing is NP-complete; (ii) a linear-time algorithm for single-source DAGs under outerface-embedding constraints; and (iii) establishing an upper bound of $2$ on the crossing number of outerpath graphs, along with a tight quadratic relationship between graph bandwidth and crossing number. These results extend classical upward planarity theory and provide new theoretical foundations for directed graph visualization and VLSI layout design.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that upward-$k$-planarity testing is NP-complete already for $k =1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward $1$-planar drawing in which all vertices are incident to the outer face.

Problem

• Studying upward k-planar drawings for DAGs with edge crossing constraints
• Analyzing crossing bounds for different graph classes like outerplanar DAGs
• Investigating computational complexity of upward k-planarity testing algorithms

Innovation

• Upward k-planar drawings with bounded edge crossings
• NP-complete upward-1-planarity testing for DAGs
• Linear time decision for single-source outer-face drawings