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 O(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 O(n 2/3 log 2/3 (n)).

We consider the algorithmic problem of finding the optimal weights and biases for a twolayer fully connected neural network to fit a given set of data points. This problem is known as empirical risk minimization in the machine learning community. We show that the problem is ∃R-complete. This complexity class can be defined as the set of algorithmic problems that are polynomial-time equivalent to finding real roots of a polynomial with integer coefficients. Our results hold even if the following restrictions are all added simultaneously.• There are exactly two output neurons.• There are exactly two input neurons.• The data has only 13 different labels.• The number of hidden neurons is a constant fraction of the number of data points.• The target training error is zero.• The ReLU activation function is used.

We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class ∀∃

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 Pupyrev [arXiv:2107.13658, 2021].

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A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane R 2 . Recognizing them is known to be ∃R-complete, i.e., as hard as solving a system of polynomial inequalities. In this note we describe a simple framework to translate ∃R-hardness reductions from the Euclidean plane R 2 to the hyperbolic plane H 2 . We apply our framework to prove that the recognition of unit disk graphs in the hyperbolic plane is also ∃R-complete.

Cops and Robber is a family of two-player games played on graphs in which one player controls a number of cops and the other player controls a robber. In alternating turns, each player moves (all) his/her figures. The cops try to capture the robber while the latter tries to flee indefinitely. In this paper we consider a variant of the game played on a planar graph where the robber moves between adjacent vertices while the cops move between adjacent faces. The cops capture the robber if they occupy all incident faces. We prove that a constant number of cops suffices to capture the robber on any planar graph of maximum degree ∆ if and only if ∆ ≤ 4.

It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edgecolorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an n-vertex planar graph, augments this graph in O(n 2 ) steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized Factor-problem for the fixed embedding case, and SPQR-trees for the variable embedding case. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms → Graph algorithms analysis