Trade-off between spread and width for tree decompositions

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 $O(tw(G))$.

💡 Research Summary

The paper investigates the fundamental trade‑off between two central parameters of a tree decomposition of a graph G: its width (the size of the largest bag minus one) and its spread (the number of bags that contain a given vertex). While the classical goal is to minimise width, recent work by Wood introduced the additional requirement that each vertex v appear in only O(d(v)) bags, where d(v) is the degree of v. Wood asked for the smallest constant c such that, for some constant c′, every graph G admits a tree decomposition of width at most c·tw(G) (tw(G) being the treewidth) in which each vertex v has spread at most c′·(d(v)+1). Wood proved an upper bound c≤14 and left the exact threshold open.

The authors settle this question up to a narrow interval. Their main results are:

• Theorem 3 (sufficiency) – For every real c > 3 there exists a constant c′ > 0 such that every graph G has a tree decomposition of width at most c·tw(G) and spread at most c′·(d(v)+1) for every vertex v. The construction builds on the notion of domino treewidth and uses a refined bag‑splitting technique: by allowing the width to be roughly three times the optimal, they can distribute each vertex’s incident edges over a bounded number of bags, achieving linear dependence on the degree.

• Theorem 4 (necessity) – For any c < 2 and any c′ > 0 there exists a graph G (with treewidth k) such that every tree decomposition of width at most c·k contains a vertex whose spread exceeds c′·(d(v)+1). The lower bound is proved via a carefully crafted family of “augmented grid” graphs D⁺_{n,m}. These graphs have treewidth Θ(n) but contain a special vertex v₀ of degree Θ(n²). The authors show that any decomposition whose width is less than 2·tw(G) must place v₀ in at least Ω(n⁴) bags, i.e., its spread is at least ½·n²·(d(v₀)+1). Consequently, a factor smaller than 2 on the width cannot guarantee degree‑proportional spread.

Together, Theorems 3 and 4 imply that the optimal constant c lies in the interval