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A linear layout of a graph consists of a linear ordering of its vertices and a partition of its edges into pages such that the edges assigned to the same page obey some constraint. The two most prominent and widely studied types of linear…

Discrete Mathematics · Computer Science 2025-08-07 Emilio Di Giacomo , Walter Didimo , Henry Förster , Torsten Ueckerdt , Johannes Zink

It is proved that the rectilinear crossing number of every graph with bounded tree-width and bounded degree is linear in the number of vertices. **** This paper has been withdrawn by the author. **** The results have been superseeded by the…

Combinatorics · Mathematics 2007-05-23 David R. Wood

A properly edge-colored graph is a graph with a coloring of its edges such that no vertex is incident to two or more edges of the same color. A subgraph is called rainbow if all its edges have different colors. The problem of finding…

Combinatorics · Mathematics 2024-12-19 Benny Sudakov

The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of…

Discrete Mathematics · Computer Science 2020-12-08 Marzieh Eidi , Amirhossein Farzam , Wilmer Leal , Areejit Samal , Jürgen Jost

For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to saturated graphs…

Computational Geometry · Computer Science 2021-08-30 Steven Chaplick , Fabian Klute , Irene Parada , Jonathan Rollin , Torsten Ueckerdt

Circular layouts are a popular graph drawing style, where vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is \NP-hard. One way to allow for fewer crossings in practice are…

Computational Geometry · Computer Science 2018-03-16 Fabian Klute , Martin Nöllenburg

Morphing edge drawing (MED), a graph drawing technique, is a dynamic extension of partial edge drawing (PED), where partially drawn edges (stubs) are repeatedly stretched and shrunk by morphing. Previous experimental evaluations have shown…

Data Structures and Algorithms · Computer Science 2022-09-01 Kazuo Misue

The activations of Facial Action Units (AUs) mutually influence one another. While the relationship between a pair of AUs can be complex and unique, existing approaches fail to specifically and explicitly represent such cues for each pair…

Computer Vision and Pattern Recognition · Computer Science 2022-08-03 Cheng Luo , Siyang Song , Weicheng Xie , Linlin Shen , Hatice Gunes

We introduce a method to embed edge-colored graphs into families of expander graphs, which generalizes a framework developed by Dragani\'c, Krivelevich, and Nenadov (2022). As an application, we show that each family of sufficiently…

Combinatorics · Mathematics 2025-01-27 Ben Lund , Chuandong Xu

The recognition of named entities in visually-rich documents (VrD-NER) plays a critical role in various real-world scenarios and applications. However, the research in VrD-NER faces three major challenges: complex document layouts,…

Computation and Language · Computer Science 2024-08-13 Yi Tu , Chong Zhang , Ya Guo , Huan Chen , Jinyang Tang , Huijia Zhu , Qi Zhang

In a \emph{fan-planar drawing} of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every $n$-vertex fan-planar drawing has at most…

Computational Geometry · Computer Science 2019-09-04 Carla Binucci , Emilio Di Giacomo , Walter Didimo , Fabrizio Montecchiani , Maurizio Patrignani , Ioannis G. Tollis

Computing the gradients of a rendering process is paramount for diverse applications in computer vision and graphics. However, accurate computation of these gradients is challenging due to discontinuities and rendering approximations,…

Computer Vision and Pattern Recognition · Computer Science 2024-07-25 Stanislav Pidhorskyi , Tomas Simon , Gabriel Schwartz , He Wen , Yaser Sheikh , Jason Saragih

We study the problem of selecting a maximum-weight subgraph of a given graph such that the subgraph can be drawn within a prescribed drawing area subject to given non-uniform vertex sizes. We develop and analyze heuristics both for the…

Computational Geometry · Computer Science 2014-09-02 Maximilian Aulbach , Martin Fink , Julian Schuhmann , Alexander Wolff

We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices…

Discrete Mathematics · Computer Science 2020-10-06 N. R. Aravind , Udit Maniyar

Unsupervised graph representation learning (UGRL) has drawn increasing research attention and achieved promising results in several graph analytic tasks. Relying on the homophily assumption, existing UGRL methods tend to smooth the learned…

Machine Learning · Computer Science 2022-11-28 Yixin Liu , Yizhen Zheng , Daokun Zhang , Vincent CS Lee , Shirui Pan

For an angle $\alpha\in (0,\pi)$, we consider plane graphs and multigraphs in which the edges are either (i) one-bend polylines with an angle $\alpha$ between the two edge segments, or (ii) circular arcs of central angle $2(\pi-\alpha)$. We…

Discrete Mathematics · Computer Science 2023-11-28 Csaba D. Tóth

Cross-spectral face recognition (CFR) refers to recognizing individuals using face images stemming from different spectral bands, such as infrared versus visible. While CFR is inherently more challenging than classical face recognition due…

Computer Vision and Pattern Recognition · Computer Science 2024-10-10 David Anghelone , Cunjian Chen , Arun Ross , Antitza Dantcheva

We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the…

Combinatorics · Mathematics 2019-05-24 Siddharth Prasad

We study crossing numbers of dense graph drawings whose vertices are uniformly distributed either on the unit sphere or in a compact convex planar domain. We prove a sharp inequality for weighted geodesic drawings on $\mathbb S^2$ in a…

Combinatorics · Mathematics 2026-05-19 Saba Lepsveridze , Oriol Solé-Pi

A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently,…

Discrete Mathematics · Computer Science 2021-07-16 Michael Kaufmann , Torsten Ueckerdt