Trade-off between spread and width for tree decompositions

TL;DR

This work analyzes the trade-off between the width and the spread of vertex occurrences in tree decompositions, addressing Wood's questions on whether a width bound of $O(tw(G))$ can accompany spread controlled by degree. It establishes that the width-spread constant must satisfy $c\ge 2$ and can be taken for all graphs when $c>3$, and it further shows that near-optimal average spread can be achieved with width $O(tw(G))$, via a constructive framework using balanced separators, slick decompositions, and $b$-markings. The authors provide grid-based lower bounds illustrating inherent spread constraints at fixed width, and they present algorithmic implications showing how to compute such decompositions with favorable width and spread properties in polynomial time. They also propose open conjectures, notably that the optimal constant is 3, and outline avenues for future work involving grid structures and connections to domino treewidth and tree-partition width.

Abstract

We study the trade-off between (average) spread and width in tree decompositions, answering several questions from Wood [arXiv:2509.01140]. The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. Wood asked for which $c>0$, there exists $c'$ such that each graph $G$ has a tree decomposition of width $c\cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that $c\geq 2$ is necessary and that $c>3$ is sufficient. Moreover, we answer a second question fully by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.

Theorems & Definitions (15)

• Theorem 2: Theorem 2 in wood2025treedecompositionssmallwidth
• Theorem 3
• Theorem 4
• Theorem 6
• Lemma 8: Folklore
• Lemma 9
• Theorem 10: Variant of Hex theorem
• Corollary 11
• Lemma 12
• Theorem 13