An efficient recursive decomposition algorithm for undirected graphs

TL;DR

This paper presents a recursive decomposition algorithm for undirected graphs based on maximum cardinality search, which simplifies complex graph problems without needing minimal triangulation, and demonstrates improved efficiency through experiments.

Contribution

It introduces a novel recursive decomposition method leveraging MCS ordering, eliminating the need for minimal triangulation and clique separator identification.

Findings

The algorithm effectively decomposes graphs into atoms.

Experimental results show significant efficiency improvements.

The method aligns with chordal graph decomposition principles.

Abstract

The decomposition of undirected graphs simplifies complex problems by breaking them into solvable subgraphs, following the philosophy of divide and conquer. This paper investigates the relationship between atom decomposition and the maximum cardinality search (MCS) ordering in general undirected graphs. Specifically, we prove that applying a convex extension to the node numbered $1$ and its neighborhood in an MCS ordering yields an atom in the graph. Furthermore, based on the MCS ordering, we introduce a recursive algorithm for decomposing an undirected graph into its atoms. This approach closely aligns with the results of chordal graph decomposition. As a result, minimal triangulation of the graph is no longer required, and the identification of clique minimal separators is avoided. In the experimental section, we combine the proposed decomposition algorithm with two existing convex expansion methods. The results show that both combinations significantly outperform the existing algorithms in terms of efficiency.