相关论文: Toward a Unit Distance Embedding for the Heawood g…
Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of…
In this paper, two results are obtained on a hypergraph embedding problem. The proof technique is itself of interest, being the first time amalgamations have been used to address the embedding of hypergraphs. The first result finds…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
Graphs are a representation of structured data that captures the relationships between sets of objects. With the ubiquity of available network data, there is increasing industrial and academic need to quickly analyze graphs with billions of…
In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of…
We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A \subseteq…
We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region…
When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in…
Embedding undirected graphs in a Euclidean space has many computational benefits. FastMap is an efficient embedding algorithm that facilitates a geometric interpretation of problems posed on undirected graphs. However, Euclidean distances…
We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…
The problem of determining whether a graph $G$ can be realized as a unit-distance graph in $\mathbb{Z}^2$ is NP-complete. As far as we can tell, a proof of this result has never been written up. We prove NP-completeness of this problem by…
In this work, we study how far one can deviate from optimal behavior when embedding a planar graph. For a planar graph $G$, we say that a plane subgraph $H\subseteq G$ is a \textit{plane-saturated subgraph} if adding any edge (possibly with…
The computation of distance measures between nodes in graphs is inefficient and does not scale to large graphs. We explore dense vector representations as an effective way to approximate the same information: we introduce a simple yet…
Planarity Testing is the problem of determining whether a given graph is planar while planar embedding is the corresponding construction problem. The bounded space complexity of these problems has been determined to be exactly Logspace by…
Trajectories that capture object movement have numerous applications, in which similarity computation between trajectories often plays a key role. Traditionally, the similarity between two trajectories is quantified by means of heuristic…
Learning low-dimensional numerical representations from symbolic data, e.g., embedding the nodes of a graph into a geometric space, is an important concept in machine learning. While embedding into Euclidean space is common, recent…
Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there…
Uniform Manifold Approximation and Projection (UMAP) is one of the state-of-the-art methods for dimensionality reduction and data visualization. This is a tutorial and survey paper on UMAP and its variants. We start with UMAP algorithm…
Statistical analysis of a graph often starts with embedding, the process of representing its nodes as points in space. How to choose the embedding dimension is a nuanced decision in practice, but in theory a notion of true dimension is…
Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored:…