Related papers: CelticGraph: Drawing Graphs as Celtic Knots and Li…
Interaction nets are a graphical formalism inspired by Linear Logic proof-nets often used for studying higher order rewriting e.g. \Beta-reduction. Traditional presentations of interaction nets are based on graph theory and rely on…
Concept lattice drawings are an important tool to visualize complex relations in data in a simple manner to human readers. Many attempts were made to transfer classical graph drawing approaches to order diagrams. Although those methods are…
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot…
A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed…
The classical knot recognition problem is the problem of determining whether the virtual knot represented by a given diagram is classical. We prove that this problem is in NP, and we give an exponential time algorithm for the problem.
In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion…
Knowledge graphs are graphical representations of large databases of facts, which typically suffer from incompleteness. Inferring missing relations (links) between entities (nodes) is the task of link prediction. A recent state-of-the-art…
In this paper, we introduce the notion of a finite non-simple directed graph, called an ornated graph and initiate a study on ornated graphs. An ornated graph is a directed graph on $n$ vertices, denoted by $O_n(s_l)$, whose vertices are…
We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide…
Many systems comprising entities in interactions can be represented as graphs, whose structure gives significant insights about how these systems work. Network theory has undergone further developments, in particular in relation to…
Knowledge graph embedding involves learning representations of entities -- the vertices of the graph -- and relations -- the edges of the graph -- such that the resulting representations encode the known factual information represented by…
We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.
Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph…
Context: Edge graphs are graphs whose edges are labelled with identifiers, and nodes can have multiple edges between them. They are used to model a wide range of systems, including networks with distances or degrees of connection and…
A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we consider several problems of graph theory and developed optimal algorithms to solve such problems on cactus graphs. The running time of…
Inspired by artistic practices such as beadwork and himmeli, we study the problem of threading a single string through a set of tubes, so that pulling the string forms a desired graph. More precisely, given a connected graph (where edges…
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…
This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing…
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…
Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general…