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arXiv · 2509.01140

Tree decompositions with small width, spread, order and degree

David R. Wood
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Abstract

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.

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David R. Wood. 2025-09-01. Tree decompositions with small width, spread, order and degree. https://arxiv.org/abs/2509.01140

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