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.

翻译：无向图的分解遵循分治思想，通过将复杂问题拆解为可求解的子图来简化问题。本文研究了一般无向图中原子分解与最大基数搜索（MCS）排序之间的关系。具体而言，我们证明了在 MCS 排序中对编号为 $1$ 的节点及其邻域应用凸扩展可得到图中的一个原子。此外，基于 MCS 排序，我们提出了一种将无向图递归分解为原子的算法。该方法与弦图分解的结果高度一致。因此，不再需要对图进行最小三角剖分，也避免了对团最小分离子的识别。在实验部分，我们将所提出的分解算法与两种现有的凸扩展方法相结合。结果表明，两种组合方案在效率上均显著优于现有算法。