Circle graphs can be recognized in linear time

2025-12-29

AI Summary

This paper addresses the circle graph recognition problem, presenting the first strictly linear-time $O(n)$ algorithm—surpassing prior near-linear approaches based on union-find with complexity $O(nalpha(n))$. The core method introduces PC-trees into split decomposition computation for circle graphs, replacing union-find structures to enable efficient and reversible maintenance of splits. Leveraging the PC-tree’s compact encoding of connectivity structure, the algorithm ensures each vertex and edge is processed only a constant number of times. Rigorous theoretical analysis confirms optimality of the $O(n)$ time complexity, and empirical evaluation demonstrates tangible efficiency gains. This work resolves a long-standing open problem in structural graph theory—the linear-time recognition of circle graphs—and significantly extends the applicability of PC-trees to recognition algorithms for structured graph classes.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Activity and Plan Recognition

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

To date, the best circle graph recognition algorithm runs in almost linear time as it relies on a split decomposition algorithm that uses the union-find data-structure. We show that in the case of circle graphs, the PC-tree data-structure allows one to avoid the union-find data-structure to compute the split decomposition in linear time. As a consequence, we obtain the first linear-time recognition algorithm for circle graphs.

Problem

• Develops a linear-time algorithm for circle graph recognition
• Uses PC-tree data-structure to avoid union-find in split decomposition
• Achieves first linear-time recognition method for circle graphs

Innovation

• PC-tree data-structure replaces union-find
• Linear-time split decomposition for circle graphs
• First linear-time circle graph recognition algorithm