Publications by Radoslav Fulek

Thrackles: An improved upper bound

A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, ex ...

ELSEVIER SCIENCE BV2019

Free Edge Lengths in Plane Graphs

Radoslav Fulek, Filip Moric

We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the "host" of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a pla ...

Springer2015

Clustered planarity testing revisited

Radoslav Fulek, Jan Kyncl, Igor Malinovic

The Hanani--Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generali ...

2015

Homometric sets in trees

Radoslav Fulek, Slobodan Mitrovic

Let G = (V, E) denote a simple graph with vertex set V and edge set E. The profile of a vertex set V' subset of V denotes the multiset of pairwise distances between the vertices of V'. Two disjoint subsets of V are homometric if their profiles are the same ...

Academic Press Ltd- Elsevier Science Ltd2014

On Polygons Excluding Point Sets

Radoslav Fulek, Filip Moric, Balázs Keszegh

By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that is in general position, and the convex hull of B c ...

Springer Japan2013

Graphs That Admit Polyline Drawings with Few Crossing Angles

Radoslav Fulek

We consider graphs that admit polyline drawings where all crossings occur at the same angle alpha is an element of (0, pi/2]. We prove that every graph on n vertices that admits such a polyline drawing with at most two bends per edge has O(n) edges. This r ...

SIAM Publications2012

Graphs that admit right angle crossing drawings

Radoslav Fulek, Filip Moric, Balázs Keszegh

Elsevier2012

Intersection Patterns of Edges in Topological Graphs

Radoslav Fulek

This thesis is devoted to crossing patterns of edges in topological graphs. We consider the following four problems: A thrackle is a graph drawn in the plane such that every pair of edges meet exactly once: either at a common endpoint or in a proper crossi ...

EPFL2012

Diameter Bounds for Planar Graphs

Radoslav Fulek, Filip Moric, David Pritchard

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method ...

Elsevier2011

A computational approach to Conway's thrackle conjecture

János Pach, Radoslav Fulek

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 yea ...

2011

Orthogeodesic Point-Set Embedding of Trees

Radoslav Fulek, Fabrizio Frati

Let S be a set of N grid points in the plane, no two of which lie on the same horizontal or vertical line, and let G be a graph with n vertices (n

Elsevier Science Bv2011

On the Page Number of Upward Planar Directed Acyclic Graphs

Radoslav Fulek, Fabrizio Frati, Andres Jacinto Ruiz Vargas

In this paper we study the page number of upward planar directed acyclic graphs. We prove that: (I) the page number of any n-vertex upward planar triangulation G whose every maximal 4-connected component has page number k is at most min {O(k log n), O(2(k) ...

Springer-Verlag Berlin2011

Hanani-Tutte and Monotone Drawings

Radoslav Fulek

2011

Adjacent Crossings Do Matter

Radoslav Fulek

In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent if they share no endpoint. For a graph G, let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) b ...

Springer-Verlag Berlin2011

Hanani-Tutte and Monotone Drawings

Radoslav Fulek

A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Toth showed that if a graph has an x-monotone drawing in which every pair of edges crosses an even nu ...

Springer-Verlag Berlin2011

On the Size of Graphs That Admit Polyline Drawings with Few Bends and Crossing Angles

Radoslav Fulek

Springer2010

A Computational Approach to Conway's Thrackle Conjecture

János Pach, Radoslav Fulek

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 yea ...

Springer2010

Radoslav Fulek

Graph Chatbot

Thrackles: An improved upper bound

Free Edge Lengths in Plane Graphs

Clustered planarity testing revisited

Homometric sets in trees

On Polygons Excluding Point Sets

Graphs That Admit Polyline Drawings with Few Crossing Angles

Graphs that admit right angle crossing drawings

Intersection Patterns of Edges in Topological Graphs

Diameter Bounds for Planar Graphs

A computational approach to Conway's thrackle conjecture

Orthogeodesic Point-Set Embedding of Trees

On the Page Number of Upward Planar Directed Acyclic Graphs

Hanani-Tutte and Monotone Drawings

Adjacent Crossings Do Matter

Hanani-Tutte and Monotone Drawings

On the Size of Graphs That Admit Polyline Drawings with Few Bends and Crossing Angles

A Computational Approach to Conway's Thrackle Conjecture

Coloring geometric hypergraph defined by an arrangement of half-planes

On polygons excluding point sets

Intersecting convex sets by rays