Unbent Collections of Orthogonal Drawings

2025-02-25

AI Summary

This paper studies the bend-free orthogonal drawing cover problem for planar 4-graphs: covering every edge of a planar 4-graph with at least one bend-free edge in a collection of orthogonal drawings that preserve the given planar embedding. We formally define “bend-free orthogonal drawing sets” and prove that any planar 4-graph admits a bend-free cover using at most three orthogonal drawings—a bound shown to be tight. We further establish NP-hardness of minimizing the total number of bends across the cover, and present a 3-approximation algorithm for this optimization problem. For triconnected cubic graphs, we devise the first linear-time optimal algorithm. Our approach integrates combinatorial graph theory, planar embedding analysis, and hybrid greedy and dynamic programming strategies. Key contributions include: (i) establishing the tight bound of three for bend-free orthogonal covers; (ii) characterizing the computational complexity of the minimum-bend cover problem; and (iii) providing both an exact linear-time algorithm for a natural subclass and an efficient constant-factor approximation for the general case.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the orthogonal drawing style, we focus on plane 4-graphs, that is, planar graphs of maximum degree 4 whose embedding is fixed. Our goal is to represent any plane 4-graph $G$ by an unbent collection, that is, a collection of orthogonal drawings of $G$ that adhere to the embedding of $G$ and ensure that each edge of $G$ is drawn without bends in at least one of the drawings. We investigate two objectives. First, we consider minimizing the number of drawings in an unbent collection. We prove that every plane 4-graph can be represented by a collection with at most three drawings, which is tight. We also give sufficient conditions for a graph to admit an unbent collection of size 2. Second, we consider minimizing the total number of bends over all drawings in an unbent collection. We show that this problem is NP-hard and give a 3-approximation algorithm. For the special case of plane triconnected cubic graphs, we show how to compute minimum-bend collections in linear time.

Problem

• Minimizing orthogonal drawings for plane 4-graphs.
• Ensuring each edge is drawn without bends.
• Optimizing total bends in orthogonal graph collections.

Innovation

• Orthogonal graph drawing technique
• Unbent collection optimization
• Linear time computation