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 product structure, i.e., that there is no constant c such that every such graph is a subgraph of H ⊠ 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].

The geodesic treewidth of a graph G is the smallest k for which there is a partition 𝒫 into geodesics such that G/𝒫 has treewidth k, where G/𝒫 is obtained from G by contracting each part of 𝒫. Based on this notion, row treewidth was developed and is defined for a graph G as the smallest k such that G ⊆ H ⊠ P for some graph H of treewidth k and a path P. Equivalently, the row treewidth of a graph G is the smallest k for which there is a partition 𝒫 into disjoint unions of geodesics that are aligned with respect to some layering such that G/𝒫 has treewidth k. We separate the two notions by showing that bounded row treewidth does not imply bounded geodesic treewidth and by presenting a polynomial-time algorithm to decide whether a graph of treewidth 2 has geodesic treewidth 1, which is known to be NP-hard for row treewidth [Biedl, Eppstein, Ueckerdt, 2025]. More generally, we provide an algorithm to decide whether a given graph has geodesic treewidth at most d that is XP in the treewidth, whereas there is no such algorithm for row treewidth, unless P = NP [Biedl, Eppstein, Ueckerdt, 2025]. On the other hand, we show that computing the geodesic treewidth is NP-hard and that every graph with geodesic treewidth 1 has bounded row treewidth. Moreover, we improve the best known lower bound on the geodesic treewidth of planar graphs to 5.

A graph class $\mathcal{G}$ admits product structure if there exists a constant $k$ such that every $G \in \mathcal{G}$ is a subgraph of $H \boxtimes P$ for a path $P$ and some graph $H$ of treewidth $k$. Famously, the class of planar graphs, as well as many beyond-planar graph classes are known to admit product structure. However, we have only few tools to prove the absence of product structure, and hence know of only a few interesting examples of classes. Motivated by the transition between product structure and no product structure, we investigate subclasses of intersection graphs in the plane (e.g., disk intersection graphs) and present necessary and sufficient conditions for these to admit product structure. Specifically, for a set $S \subset \mathbb{R}^2$ (e.g., a disk) and a real number $α\in [0,1]$, we consider intersection graphs of $α$-free homothetic copies of $S$. That is, each vertex $v$ is a homothetic copy of $S$ of which at least an $α$-portion is not covered by other vertices, and there is an edge between $u$ and $v$ if and only if $u \cap v \neq \emptyset$. For $α= 1$ we have contact graphs, which are in most cases planar, and hence admit product structure. For $α= 0$ we have (among others) all complete graphs, and hence no product structure. In general, there is a threshold value $α^*(S) \in [0,1]$ such that $α$-free homothetic copies of $S$ admit product structure for all $α> α^*(S)$ and do not admit product structure for all $α< α^*(S)$. We show for a large family of sets $S$, including all triangles and all trapezoids, that it holds $α^*(S) = 1$, i.e., we have no product structure, except for the contact graphs (when $α= 1$). For other sets $S$, including regular $n$-gons for infinitely many values of $n$, we show that $0 < α^*(S) < 1$ by proving upper and lower bounds.

The stack number of a directed acyclic graph G is the minimum k for which there is a topological ordering of G and a k-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [GD 2021] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed H-partitions, which might be of independent interest.We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by Nöllenburg and Pupyrev [GD 2023].

Abstract. The page number of a directed acyclic graph [Formula: see text] is the minimum [Formula: see text] for which there is a topological ordering of [Formula: see text] and a [Formula: see text]-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We address the long-standing open problem asking for the largest page number among all upward planar graphs. We improve the best known lower bound to 5 and present the first asymptotic improvement over the trivial [Formula: see text] upper bound, where [Formula: see text] denotes the number of vertices in [Formula: see text]. Specifically, we first prove that the page number of every upward planar graph is bounded in terms of its width, as well as its height. We then combine both approaches to show that every [Formula: see text]-vertex upward planar graph has page number [Formula: see text].

The page number of a directed acyclic graph G is the minimum k for which there is a topological ordering of G and a k-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We address the long-standing open problem asking for the largest page number among all upward planar graphs. We improve the best known lower bound to 5 and present the first asymptotic improvement over the trivial (n) upper bound, where n denotes the number of vertices in G. Specifically, we first prove that the page number of every upward planar graph is bounded in terms of its width, as well as its height. We then combine both approaches to show that every n-vertex upward planar graph has page number (n2/3 log2/3(n)).