Tree-decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. The main property of tree-decompositions is the width (the maximum size of a bag minus 1). We show that every graph has a tree-decomposition with near-optimal width, where each vertex appears in few bags. In particular, every graph with treewidth $k$ has a tree-decomposition with width at most $14k+13$, where each vertex $v$ appears in at most $\text{deg}(v)+1$ bags. This improves an exponential bound by Ding and Oporowski [1995] to linear, and establishes a conjecture of theirs in a strong sense. In a second result, we show that every graph with treewidth $k$ has a tree-decomposition with width at most $3k-1$, where on average each vertex appears in at most three bags.

翻译：图的树分解在结构图论和算法图论中具有基础重要性。树分解的主要属性是宽度（一个袋子的最大尺寸减1）。我们证明：每个图都存在具有接近最优宽度的树分解，且每个顶点出现在少数袋子中。特别地，每个树宽为$k$的图都存在一个宽度至多为$14k+13$的树分解，其中每个顶点$v$出现在至多$\text{deg}(v)+1$个袋子中。这比Ding与Oporowski [1995]的指数界改进为线性界，并在强意义下验证了他们提出的一个猜想。在第二个结论中，我们证明：每个树宽为$k$的图都存在一个宽度至多为$3k-1$的树分解，其中每个顶点平均出现在至多三个袋子中。