分裂分解与图标号树：完全可分解图的刻画与全动态算法
Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs
摘要：
在本文中，我们重新审视分裂分解的图，并给出新的组合和算法的结果，一类完全可分解的图，也被称为距离遗传图，并为两个非平凡的子类，即cographs和3叶功率图。准确地说，我们给结构和增量表征，导致最佳的全动态识别算法的顶点和边缘修改，为这些类。这些结果依赖于一个新的框架来表示分裂分解，即图标记树，它也捕获了图的模块化分解，从而统一了这两种分解技术。本文的重点是使用这些图形类和树的节点标记的集团和明星之间的双射。这样做，我们也能够得到一个距离遗传图的交集模型，它回答了一个公开的问题。
英文摘要：
In this paper, we revisit the split decomposition of graphs and give new combinatorial and algorithmic results for the class of totally decomposable graphs, also known as the distance hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give strutural and incremental characterizations, leading to optimal fully-dynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on a new framework to represent the split decomposition, namely the graph-labelled trees, which also captures the modular decomposition of graphs and thereby unify these two decompositions techniques. The point of the paper is to use bijections between these graph classes and trees whose nodes are labelled by cliques and stars. Doing so, we are also able to derive an intersection model for distance hereditary graphs, which answers an open problem.
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