基于图标记树的实用高效分裂分解
Practical and Efficient Split Decomposition via Graph-Labelled Trees
摘要：
图的分裂分解是由坎宁安（Cunningham）引入的（以连接分解的名义），作为模分解的推广。本文对分裂分解的算法性质进行了研究。我们这样做的背景下，图标记树（GLT），一个新的组合对象，旨在简化其考虑。GLT是用来获得一个增量表征分裂分解，与一个简单的组合描述，并探讨其属性相对于词典宽度优先搜索（LBFS）。将增量表征应用于LBFS排序导致在时间上运行的分裂分解算法$O(n+m)α(n+m)$，其中$α$是逆阿克曼函数，其值对于任何实际图都小于4。相比Dahlhaus的线性时间分裂分解算法[Dahlhaus'00]，它不依赖于一个增量的建设，我们的算法是一样快，但本文给出的渐近意义和完整的实现细节。此外，我们的算法扩展到圆图识别，而没有这样的扩展是已知的Dahlhaus算法。配套文件[Gioan等人]使用我们的算法推导出第一个次二次圆图识别算法。
英文摘要：
Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time $O(n+m)α(n+m)$, where $α$ is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.] uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm.
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