Related papers: Dynamic Planar Embeddings of Dynamic Graphs
Planar Embedding is a drawing of a graph on the plane such that the edges do not intersect each other except at the vertices. We know that testing the planarity of a graph and computing its embedding (if it exists), can efficiently be…
Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider…
A directed graph $G$ is upward planar if it admits a planar embedding such that each edge is $y$-monotone. Unlike planarity testing, upward planarity testing is NP-hard except in restricted cases, such as when the graph has the…
We study dynamic planar graphs with $n$ vertices, subject to edge deletion, edge contraction, edge insertion across a face, and the splitting of a vertex in specified corners. We dynamically maintain a combinatorial embedding of such a…
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…
Dynamic graphs refer to graphs whose structure dynamically changes over time. Despite the benefits of learning vertex representations (i.e., embeddings) for dynamic graphs, existing works merely view a dynamic graph as a sequence of changes…
Dynamic Connectivity is a fundamental algorithmic graph problem, motivated by a wide range of applications to social and communication networks and used as a building block in various other algorithms, such as the bi-connectivity and the…
Embedding static graphs in low-dimensional vector spaces plays a key role in network analytics and inference, supporting applications like node classification, link prediction, and graph visualization. However, many real-world networks…
In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes. We investigate whether a given upward planar digraph admits such a…
Given a dynamic graph subject to insertions and deletions of edges, a natural question is whether the graph presently admits a planar embedding. We give a deterministic fully-dynamic algorithm for general graphs, running in amortized…
We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and…
We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form `Are vertices $u$ and $v$ connected with a path?' in…
We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An…
Any planar graph has a crossing-free straight-line drawing in the plane. A simultaneous geometric embedding of two n-vertex graphs is a straight-line drawing of both graphs on a common set of n points, such that the edges withing each…
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…
We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time…
Consider two planar graphs which are subject to edge insertions and deletions. We show that whether the two graphs are isomorphic can be maintained with first-order logic formulas and auxiliary data of polynomial size. This places the…
A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to…
Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges…
We study dynamic graph algorithms in the Massively Parallel Computation model, which was inspired by practical data processing systems. Our goal is to provide algorithms that can efficiently handle large batches of edge insertions and…