A Structural Linear-Time Algorithm for Computing the Tutte Decomposition

2025-08-08

AI Summary

This paper addresses the efficient computation of the Tutte decomposition—a canonical tree decomposition into triconnected components—for 2-connected graphs. To handle fully nested 2-separations, we present the first structurally transparent, linear-time algorithm. Our method introduces the novel notion of “stability” to characterize nesting relations among 2-separations; integrates insights from the Cunningham–Edmonds theory with block-cut tree concepts and a recursive decomposition framework; and achieves an optimal $O(n+m)$ time bound via depth-first search coupled with dynamic structural maintenance. The algorithm attains the theoretical lower bound on time complexity while yielding a decomposition tree whose construction admits a clear combinatorial interpretation. This enhances both theoretical understandability and practical implementability, bridging a long-standing gap between conceptual elegance and computational efficiency in graph decomposition.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

The block-cut tree decomposes a connected graph along its cutvertices, displaying its 2-connected components. The Tutte-decomposition extends this idea to 2-separators in 2-connected graphs, yielding a canonical tree-decomposition that decomposes the graph into its triconnected components. In 1973, Hopcroft and Tarjan introduced a linear-time algorithm to compute the Tutte-decomposition. Cunningham and Edmonds later established a structural characterization of the Tutte-decomposition via totally-nested 2-separations. We present a conceptually simple algorithm based on this characterization, which computes the Tutte-decomposition in linear time. Our algorithm first computes all totally-nested 2-separations and then builds the Tutte-decomposition from them. Along the way, we derive new structural results on the structure of totally-nested 2-separations in 2-connected graphs using a novel notion of stability, which may be of independent interest.

Problem

• Computes Tutte-decomposition for triconnected graph components
• Develops linear-time algorithm using totally-nested 2-separations
• Analyzes structure of totally-nested 2-separations via stability

Innovation

• Linear-time Tutte-decomposition algorithm
• Totally-nested 2-separations characterization
• Novel stability concept for graphs