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The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological…
An orthogonal representation of a graph $G$ over a field $\mathbb{F}$ is an assignment of a vector $u_v \in \mathbb{F}^t$ to every vertex $v$ of $G$, such that $\langle u_v,u_v \rangle \neq 0$ for every vertex $v$ and $\langle u_v,u_{v'}…
Graph is an important data representation which appears in a wide diversity of real-world scenarios. Effective graph analytics provides users a deeper understanding of what is behind the data, and thus can benefit a lot of useful…
We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world…
Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs…
Given a planar graph $G$ and an integer $b$, OrthogonalPlanarity is the problem of deciding whether $G$ admits an orthogonal drawing with at most $b$ bends in total. We show that OrthogonalPlanarity can be solved in polynomial time if $G$…
Low-dimensional representations, or embeddings, of a graph's nodes facilitate several practical data science and data engineering tasks. As such embeddings rely, explicitly or implicitly, on a similarity measure among nodes, they require…
A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…
We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional…
Graph embeddings have become a key and widely used technique within the field of graph mining, proving to be successful across a broad range of domains including social, citation, transportation and biological. Graph embedding techniques…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
We propose a new method for embedding graphs while preserving directed edge information. Learning such continuous-space vector representations (or embeddings) of nodes in a graph is an important first step for using network information…
Graph embedding is an effective method to represent graph data in a low dimensional space for graph analytics. Most existing embedding algorithms typically focus on preserving the topological structure or minimizing the reconstruction…
Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As…
A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the…
Symmetry is an important aesthetic criteria in graph drawing and network visualisation. Symmetric graph drawings aim to faithfully represent automorphisms of graphs as geometric symmetries in a drawing. In this paper, we design and…
We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge…
We consider embeddings of planar graphs in $R^2$ where vertices map to points and edges map to polylines. We refer to such an embedding as a polyline drawing, and ask how few bends are required to form such a drawing for an arbitrary planar…
In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these…
The vertex connectivity of a graph $G$ is the size of the smallest set of vertices $S$ such that $G \setminus S$ is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and…