Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs

Abstract

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].

Figures (13)

• Figure 1: HUDGs with random vertex positions. Left: The radius $r$ is so small (think of $1/\sqrt{n}$) that it is indistinguishable from the Euclidean setting. Right: The disk radius $r$ is logarithmic in $n$.
• Figure 2: The strong product of two paths (left), of a graph $H$ and a clique (middle), and a graph $H$ with a path (right)
• Figure 3: Left: Strong product of a graph $H$ (red) with a path $P$ (blue). The three different types of edges are color coded in red (edge in $H$, same vertex in $P$), blue (edge in $P$, same vertex in $H$), and gray (edge in $H$ and $P$). Right: Strong product of a graph (red) with a $K_3$ (blue).
• Figure 4: Regular tilings (top) together with their duals (bottom). From left to right: A Euclidean $\{6, 3\}$ tiling, a hyperbolic $\{7, 3\}$ tiling, and a hyperbolic $\{4, 5\}$-tiling. The hyperbolic tilings are shown in the Poincaré disk. While the tiles seem to get smaller towards the boundary of the disk, they are all congruent.
• Figure 5: Left: Vertex positions of $G_r$ constructed for \ref{['thm:hypergrid']}, where the $k$-th level, $k \geq 2$, induces a $2^k$-cycle with consecutive vertices having angular distance $2\pi / 2^k$. Center: Equivalent construction inspired by the separators in blasius_hyperbolic_2016, where the vertices are placed at intersection points of hypercycles. Right: A right triangle for $A_k$ justifying \ref{['obs:sinh_rk']}.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Lemma 6
• Lemma 7: dujmovic_layered_2017, dujmovic_planar_2019, see also bose_ltw_rtw_2022
• Theorem 8
• Theorem 9
• Lemma 11