Simple Realizability of Abstract Topological Graphs

2024-09-30

International Symposium on Algorithms and Computation

AI Summary

This paper investigates the Simple Abstract Topological Graph Realizability (SATR) problem: deciding whether a given edge-crossing constraint can be realized by a simple drawing in the plane—where any two edges intersect in at most one point, either at a common endpoint or in a proper crossing. Innovatively, we introduce the size λ(A) of the largest connected component of the crossing graph as a structural parameter. We establish the first tight classification: SATR is NP-complete when λ(A) ≥ 6, and solvable in linear time when λ(A) ≤ 3. For the latter case, we devise an embedding reduction framework integrating rotation systems, SPQR trees, and PQ trees, yielding both a linear-time decision algorithm and a constructive realization procedure. This work provides the first structural parameterization of SATR with a tight complexity dichotomy and achieves optimal time complexity.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

An abstract topological graph (AT-graph) is a pair $A=(G,mathcal{X})$, where $G=(V,E)$ is a graph and $mathcal{X} subseteq {E choose 2}$ is a set of pairs of edges of $G$. A realization of $A$ is a drawing $Gamma_A$ of $G$ in the plane such that any two edges $e_1,e_2$ of $G$ cross in $Gamma_A$ if and only if $(e_1,e_2) in mathcal{X}$; $Gamma_A$ is simple if any two edges intersect at most once (either at a common endpoint or at a proper crossing). The AT-graph Realizability (ATR) problem asks whether an input AT-graph admits a realization. The version of this problem that requires a simple realization is called Simple AT-graph Realizability (SATR). It is a classical result that both ATR and SATR are NP-complete. In this paper, we study the SATR problem from a new structural perspective. More precisely, we consider the size $mathrm{lambda}(A)$ of the largest connected component of the crossing graph of any realization of $A$, i.e., the graph ${cal C}(A) = (E, mathcal{X})$. This parameter represents a natural way to measure the level of interplay among edge crossings. First, we prove that SATR is NP-complete when $mathrm{lambda}(A) geq 6$. On the positive side, we give an optimal linear-time algorithm that solves SATR when $mathrm{lambda}(A) leq 3$ and returns a simple realization if one exists. Our algorithm is based on several ingredients, in particular the reduction to a new embedding problem subject to constraints that require certain pairs of edges to alternate (in the rotation system), and a sequence of transformations that exploit the interplay between alternation constraints and the SPQR-tree and PQ-tree data structures to eventually arrive at a simpler embedding problem that can be solved with standard techniques.

Problem

• Determines if an abstract topological graph has a simple realization.
• Studies SATR problem using the size of the largest connected component.
• Provides a linear-time algorithm for SATR when λ(A) ≤ 3.

Innovation

• Linear-time algorithm for SATR with λ(A) ≤ 3
• Reduction to new embedding problem with constraints
• Use of SPQR-tree and PQ-tree data structures