一种区分缠结的树分解的改进
Refining a Tree-Decomposition which Distinguishes Tangles
摘要：
Roberston和Seymour引入了$k$阶缠结作为表示图的高度连通部分的对象，并证明了每个图都允许粘附$<k$的树分解，其中每个$k$阶缠结都包含在不同的部分中。最近，Carmesin，Diestel，Hamann和Hundertmark证明了这样的树分解可以用一种规范的方式构造，这使得它在图的自同构下是不变的。这些规范树分解必然具有不包含$k$阶缠结的部分。我们称这些部分为不必要的。迪斯特尔问，对于那些无关紧要的部分的结构可以说些什么。在本文中，我们表明，躯干的不重要的部分，在这些树分解的分支宽度$<k$，使我们能够进一步细化的规范树分解，也表明，一个类似的结果成立$k$块。
英文摘要：
Roberston and Seymour introduced tangles of order $k$ as objects representing highly connected parts of a graph and showed that every graph admits a tree-decomposition of adhesion $<k$ in which each tangle of order $k$ is contained in a different part. Recently, Carmesin, Diestel, Hamann and Hundertmark showed that such a tree-decomposition can be constructed in a canonical way, which makes it invariant under automorphisms of the graph. These canonical tree-decompositions necessarily have parts which contain no tangle of order $k$. We call these parts inessential. Diestel asked what could be said about the structure of the inessential parts. In this paper we show that the torsos of the inessential parts in these tree-decompositions have branch width $<k$, allowing us to further refine the canonical tree-decompositions, and also show that a similar result holds for $k$-blocks.
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