In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan (2015) as a possible edge-version of tree decompositions. We show that every graph admits a tree-cut decomposition of minimum width that additionally satisfies an edge-connectivity condition analogous to Thomas' leanness.

中文翻译：

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树切割宽度的类似Menger的属性

在1990年，Thomas证明每个图都接受最小宽度的树分解，该分解还满足某种称为leanness的顶点连通性条件。该结果有许多用途，并已扩展到其他几种分解。在本文中，我们考虑了Wollan（2015）引入的树形分解作为树形分解的可能边沿版本。我们表明，每张图都接受最小宽度的树形分解，该分解还满足类似于托马斯的倾斜度的边连接条件。