🤖 AI Summary
This work proposes an efficient method for atomic decomposition of undirected graphs that circumvents the complex pipeline traditionally reliant on minimal triangulation and minimal clique separators. For the first time, it establishes a theoretical connection between Maximum Cardinality Search (MCS) orderings and atomic decomposition, enabling the direct recursive generation of atoms through the convex extension of node 1 and its neighborhood. By integrating two existing convex extension strategies, the proposed algorithm achieves significantly improved computational efficiency while preserving correctness. Experimental results demonstrate that the method outperforms current state-of-the-art graph decomposition algorithms in terms of runtime performance.
📝 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.