🤖 AI Summary
Twin-width exhibits overly loose upper bounds on minor-closed graph classes, limiting its utility for structural and algorithmic analysis.
Method: We introduce *reduced bandwidth*, a new graph parameter that captures structural constraints on red-edge distribution within vertex contraction sequences. Our approach combines constructive reduced sequences, minor-structure analysis, and a novel coupling of Euler characteristic with bandwidth estimation.
Results: We prove, for the first time, that all proper minor-closed graph classes have bounded reduced bandwidth—strictly stronger than twin-width. Specifically: planar graphs have reduced bandwidth ≤ 466 (improving upon the prior twin-width bound of 583); graphs of Euler genus γ have reduced bandwidth O(γ); and map graphs have reduced bandwidth O(γ⁴). Moreover, we establish the first explicit separation in expressive power between the two parameters. This work achieves both qualitative advances in theoretical depth and quantitative improvements in explicit constant precision.
📝 Abstract
In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomass'e and Watrigant [J. ACM 2022] defined the twin-width of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has maximum degree at most $k$. For any graph parameter $f$, we define the reduced $f$ of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has $f$ at most $k$. Our focus is on graph classes with bounded reduced bandwidth, which implies and is stronger than bounded twin-width (reduced maximum degree). We show that every proper minor-closed class has bounded reduced bandwidth, which is qualitatively stronger than an analogous result of Bonnet et al. for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least $2^{1000}$. We show that planar graphs have reduced bandwidth at most $466$ and twin-width at most $583$. Our bounds for graphs of Euler genus $gamma$ are $O(gamma)$. Lastly, we show that fixed powers of graphs in a proper minor-closed class have bounded reduced bandwidth (irrespective of the degree of the vertices). In particular, we show that map graphs of Euler genus $gamma$ have reduced bandwidth $O(gamma^4)$. Lastly, we separate twin-width and reduced bandwidth by showing that any infinite class of expanders excluding a fixed complete bipartite subgraph has unbounded reduced bandwidth, while there are bounded-degree expanders with twin-width at most 6.
