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We survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries.
We present sweeping line graphs, a generalization of $\Theta$-graphs. We show that these graphs are spanners of the complete graph, as well as of the visibility graph when line segment constraints or polygonal obstacles are considered. Our…
Let $P$ be a simple polygon with $n$ vertices, and let $A$ be a set of $m$ points or line segments inside $P$. We develop data structures that can efficiently count the number of objects from $A$ that are visible to a query point or a query…
We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can…
Inclusion and accessibility in visualization research have gained increasing attention in recent years. However, many challenges still remain to be solved on the road toward a more inclusive, shared-experience-driven visualization design…
In an undirected connected graph G=(V,E), the vertex separator problem (VSP) asks for a partition of V into nonempty subsets A, B, C such that |C| is minimized such that there is no edge between A and B, and sizes of A and B are similar.…
The Densest Subgraph Problem requires to find, in a given graph, a subset of vertices whose induced subgraph maximizes a measure of density. The problem has received a great deal of attention in the algorithmic literature since the early…
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular,…
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…
Understanding and evaluating uncertainty play a key role in decision-making. When a viewer studies a visualization that demands inference, it is necessary that uncertainty is portrayed in it. This paper showcases the importance of…
In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show…
A kind of fixed-point problem in the area of discrete tomography is proposed and investigated. Our chief concern in this paper is the case of square windows in the plane. Dealing with the arrays which are bounded, of polynomial growth, and…
We introduce the notion of watching systems in graphs, which is a generalization of that of identifying codes. We give some basic properties of watching systems, an upper bound on the minimum size of a watching system, and results on the…
We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.
When drawing graphs whose edges and nodes contain text or graphics, such informa tion needs to be displayed without overlaps, either as part of the initial layout or as a post-processing step. The core problem in removing overlaps lies in…
With the explosive growth of artificial intelligence (AI) and big data, it has become vitally important to organize and represent the enormous volume of knowledge appropriately. As graph data, knowledge graphs accumulate and convey…
Graphs are by nature unifying abstractions that can leverage interconnectedness to represent, explore, predict, and explain real- and digital-world phenomena. Although real users and consumers of graph instances and graph workloads…
We describe constructions of infinite graphs which are not representable as integral graphs in the plane, addressing a question of Erd\H{o}s. We also mention some related problems.
In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an…
This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces…