We investigate the computational complexity of the following problem. Given a “mixed” planar graph, i.e., a planar graph where some edges are directed, orient the remaining edges in such a way that the whole graph is acyclic and admits a bimodal planar embedding, that is, a planar embedding where edges entering (exiting) each vertex appear consecutively in its circular adjacency list. We show t...

We present a new algorithm to test whether a given graph G is planar and to compute a planar embedding Ĝ of G if such an embedding exists. Our algorithm utilizes a fundamentally new approach based on graph separators to obtain such an embedding. The I/O-complexity of our algorithm is O(sort(N)). A simple simulation technique reduces the I/O-complexity of our algorithm to O(perm(N)). We prove a ...

Knowledge Graph Embeddings, i.e., projections of entities and relations to lower dimensional spaces, have been proposed for two purposes: (1) providing an encoding data mining tasks, (2) predicting links in a knowledge graph. Both lines research pursued rather isolation from each other so far, with their own benchmarks evaluation methodologies. In this paper, we argue that both tasks are actual...

We present V2V, a method for embedding each vertex in a graph as a vector in a fixed dimensional space. Inspired by methods for word embedding such as word2vec, a vertex embedding is computed through enumerating random walks in the graph, and using the resulting vertex sequences to provide the context for each vertex. This embedding allows one to use well-developed techniques from machine learn...

We introduce a special type of graph embedding called book embedding and apply it to posets. A book embedding scheme for bipartite graphs is given, and is used to extend the embeddings to general k-partite graphs. Finally, we view the Hasse diagram of a poset as a directed k-partite graph and use this scheme to derive a book embedding for arbitrary posets. Using this book embedding scheme, we a...

A greedy embedding of a graph G = (V, E) into a metric space (X, d) is a function x : V (G) → X such that in the embedding for every pair of non-adjacent vertices x(s), x(t) there exists another vertex x(u) adjacent to x(s) which is closer to x(t) than x(s). This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that ever...

In this paper, we model the star interconnection network with a graph and present an innovative grid embedding into it. The embedding is speciically designed and optimized for image analysis solutions. Using the embedding, we outline the general approach for solving such problems on the star graph and analyze the data communication patterns involved in such solutions.

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a “simpler” metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, into the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-d...

Graph embedding techniques for bounding eigenvalues of associated matrices have a wide range of applications. The bounds produced by these techniques are not in general tight, however, and may be off by a log n factor for some graphs. Guattery and Miller showed that, by adding edge directions to the graph representation, they could construct an embedding called the current flow embedding, which...