arXiv subjects

Paul Jungeblut

Publications and source records attributed to Paul Jungeblut.

13 recordsLinked to original sources

arXiv2025

Directed Acyclic Outerplanar Graphs Have Constant Stack Number

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. [Eur. J. Comb., 2023] 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].

math.CO↗

arXiv2024

Recognition Complexity of Subgraphs of k-Connected Planar Cubic Graphs

We study the recognition complexity of subgraphs of k-connected planar cubic graphs for k = 1, 2, 3. We present polynomial-time algorithms to recognize subgraphs of 1- and 2-connected planar cubic graphs, both in the variable and fixed embedding setting. The main tools involve the Generalized (Anti)factor-problem for the fixed embedding case, and SPQR-trees for the variable embedding case. Secondly, we prove NP-hardness of recognizing subgraphs of 3-connected planar cubic graphs in the variable embedding setting.

cs.DM↗

arXiv2024

Cops and Robber -- When Capturing is not Surrounding

We consider "surrounding" versions of the classic Cops and Robber game. The game is played on a connected graph in which two players, one controlling a number of cops and the other controlling a robber, take alternating turns. In a turn, each player may move each of their pieces: The robber always moves between adjacent vertices. Regarding the moves of the cops we distinguish four versions that differ in whether the cops are on the vertices or the edges of the graph and whether the robber may move on/through them. The goal of the cops is to surround the robber, i.e., occupying all neighbors (vertex version) or incident edges (edge version) of the robber's current vertex. In contrast, the robber tries to avoid being surrounded indefinitely. Given a graph, the so-called cop number denotes the minimum number of cops required to eventually surround the robber. We relate the different cop numbers of these versions and prove that none of them is bounded by a function of the classical cop number and the maximum degree of the graph, thereby refuting a conjecture by Crytser, Komarov and Mackey [Graphs and Combinatorics, 2020].

math.CO↗

arXiv2024

Training Fully Connected Neural Networks is $\exists\mathbb{R}$-Complete

We consider the problem of finding weights and biases for a two-layer fully connected neural network to fit a given set of data points as well as possible, also known as EmpiricalRiskMinimization. Our main result is that the associated decision problem is $\exists\mathbb{R}$-complete, that is, polynomial-time equivalent to determining whether a multivariate polynomial with integer coefficients has any real roots. Furthermore, we prove that algebraic numbers of arbitrarily large degree are required as weights to be able to train some instances to optimality, even if all data points are rational. Our result already applies to fully connected instances with two inputs, two outputs, and one hidden layer of ReLU neurons. Thereby, we strengthen a result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021]. A consequence of this is that a combinatorial search algorithm like the one by Arora, Basu, Mianjy and Mukherjee [ICLR 2018] is impossible for networks with more than one output dimension, unless $\mathsf{NP}=\exists\mathbb{R}$.

cs.CC↗

arXiv2024

Primal-Dual Cops and Robber

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) their 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 $Δ$ if and only if $Δ\leq 4$.

math.CO↗

arXiv2023

On the Complexity of Lombardi Graph Drawing

In a Lombardi drawing of a graph the vertices are drawn as points and the edges are drawn as circular arcs connecting their respective endpoints. Additionally, all vertices have perfect angular resolution, i.e., all angles incident to a vertex $v$ have size $2π/\mathrm{deg}(v)$. We prove that it is $\exists\mathbb{R}$-complete to determine whether a given graph admits a Lombardi drawing respecting a fixed cyclic ordering of the incident edges around each vertex. In particular, this implies NP-hardness. While most previous work studied the (non-)existence of Lombardi drawings for different graph classes, our result is the first on the computational complexity of finding Lombardi drawings of general graphs.

cs.CG↗

arXiv2023

A Sublinear Bound on the Page Number of Upward Planar Graphs

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 $O(n^{2/3} \log(n)^{2/3})$.

math.CO↗

arXiv2023

Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete

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

cs.CG↗

arXiv2022

The Complexity of the Hausdorff Distance

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 $\forall\exists_<\mathbb{R}$. This implies that the problem is NP-, co-NP-, $\exists\mathbb{R}$- and $\forall\mathbb{R}$-hard.

cs.CG↗

arXiv2022

Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs

It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable 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 Antifactor-problem for the fixed embedding case, and SPQR-trees for the variable embedding case.

cs.DS↗

arXiv2022

The product structure of squaregraphs

A squaregraph is a plane graph in which each internal face is a $4$-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semi-strong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that this is best possible in the sense that "outerplanar graph" cannot be replaced by "forest".

math.CO↗

arXiv2020

Guarding Quadrangulations and Stacked Triangulations with Edges

Let $G = (V,E)$ be a plane graph. A face $f$ of $G$ is guarded by an edge $vw \in E$ if at least one vertex from $\{v,w\}$ is on the boundary of $f$. For a planar graph class $\mathcal{G}$ we ask for the minimal number of edges needed to guard all faces of any $n$-vertex graph in $\mathcal{G}$. We prove that $\lfloor n/3 \rfloor$ edges are always sufficient for quadrangulations and give a construction where $\lfloor (n-2)/4 \rfloor$ edges are necessary. For $2$-degenerate quadrangulations we improve this to a tight upper bound of $\lfloor n/4 \rfloor$ edges. We further prove that $\lfloor 2n/7 \rfloor$ edges are always sufficient for stacked triangulations (that are the $3$-degenerate triangulations) and show that this is best possible up to a small additive constant.

cs.DM↗

arXiv2018

Multilevel Planarity

In this paper, we introduce and study the multilevel-planarity testing problem, which is a generalization of upward planarity and level planarity. Let $G = (V, E)$ be a directed graph and let $\ell: V \to \mathcal P(\mathbb Z)$ be a function that assigns a finite set of integers to each vertex. A multilevel-planar drawing of $G$ is a planar drawing of $G$ such that the $y$-coordinate of each vertex $v \in V$ is $y(v) \in \ell(v)$, and each edge is drawn as a strictly $y$-monotone curve. We present linear-time algorithms for testing multilevel planarity of embedded graphs with a single source and of oriented cycles. Complementing these algorithmic results, we show that multilevel-planarity testing is NP-complete even in very restricted cases.

cs.DS↗