Hyperbolic uniform disk graphs (HUDGs) are intersection graphs of disks with some radius $r$ in the hyperbolic plane, where $r$ may be constant or depend on the number of vertices in a family of HUDGs. We show that HUDGs with constant clique number do not admit \emph{product structure}, i.e., that there is no constant $c$ such that every such graph is a subgraph of $H \boxtimes P$ for some graph $H$ of treewidth at most $c$. This justifies that HUDGs are described as not having a grid-like structure in the literature, and is in contrast to unit disk graphs in the Euclidean plane, whose grid-like structure is evident from the fact that they are subgraphs of the strong product of two paths and a clique of constant size [Dvořák et al., '21, MATRIX Annals]. By allowing $H$ to be any graph of constant treewidth instead of a path-like graph, we reject the possibility of a grid-like structure not merely by the maximum degree (which is unbounded for HUDGs) but due to their global structure. We complement this by showing that for every (sub-)constant $r$, HUDGs admit product structure, whereas the typical hyperbolic behavior is observed if $r$ grows with the number of vertices. Our proof involves a family of $n$-vertex HUDGs with radius $\log n$ that has bounded clique number but unbounded treewidth, and one for which the ratio of treewidth and clique number is $\log n / \log \log n$. Up to a $\log \log n$ factor, this negatively answers a question raised by Bläsius et al. [SoCG '25] asking whether balanced separators of HUDGs with radius $\log n$ can be covered by less than $\log n$ cliques. Our results also imply that the local and layered tree-independence number of HUDGs are both unbounded, answering an open question of Dallard et al. [arXiv '25].

翻译：双曲均匀圆盘图（HUDGs）是双曲平面上半径为 $r$ 的圆盘的交图，其中 $r$ 可以是常数，也可依赖于 HUDGs 族中顶点的数量。我们证明，团数有界的 HUDGs 不具有*产品结构*，即不存在常数 $c$ 使得每个这样的图都是某个树宽至多为 $c$ 的图 $H$ 与路径 $P$ 的强乘积 $H \boxtimes P$ 的子图。这证实了文献中描述 HUDGs 不具有网格状结构的结论，与欧几里得平面中的单位圆盘图形成对比——后者的网格状结构显而易见，因为它们是两条路径与常数大小团强乘积的子图 [Dvořák 等人，'21，MATRIX Annals]。通过允许 $H$ 为任意常数树宽图（而非路径状图），我们否定了网格状结构存在的可能性，这不仅是因为最大度（HUDGs 中的最大度无界），更是由于它们的全局结构。作为补充，我们证明对于每个（亚）常数 $r$，HUDGs 都具有产品结构，而典型的双曲行为仅在 $r$ 随顶点数增长时显现。我们的证明涉及一族半径为 $\log n$ 的 $n$ 顶点 HUDGs，其团数有界但树宽无界，且其中一个图的树宽与团数之比为 $\log n / \log \log n$。这（忽略 $\log \log n$ 因子）否定了 Bläsius 等人 [SoCG '25] 提出的问题：半径为 $\log n$ 的 HUDGs 的平衡分离器是否可以被少于 $\log n$ 个团覆盖。我们的结果还表明，HUDGs 的局部树独立数和分层树独立数均无界，解答了 Dallard 等人 [arXiv '25] 的一个开放问题。