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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…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we…
The metric sketching problem is defined as follows. Given a metric on $n$ points, and $\epsilon>0$, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up…
Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known…
We consider the problem of drawing an outerplanar graph with $n$ vertices with at most one bend per edge if the outer face is already drawn as a simple polygon. We prove that it can be decided in $O(nm)$ time if such a drawing exists, where…
In the Graph Reconstruction (GR) problem, the goal is to recover a hidden graph by utilizing some oracle that provides limited access to the structure of the graph. The interest is in characterizing how strong different oracles are when the…
We study graph ordering problems with a min-max objective. A classical problem of this type is cutwidth, where given a graph we want to order its vertices such that the number of edges crossing any point is minimized. We give a $…
An ordered graph is a graph enhanced with a linear order on the vertex set. An ordered graph is a core if it does not have an order-preserving homomorphism to a proper subgraph. We say that $H$ is the core of $G$ if (i) $H$ is a core, (ii)…
The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. Kaplan, Shamir, and Tarjan [FOCS 1994] have shown that the problem is solvable in time O(2^(O(k)) + k2 * nm) on graphs with n vertices and m…
We study the classical problem of computing geometric thickness, i.e., finding a straight-line drawing of an input graph and a partition of its edges into as few parts as possible so that each part is crossing-free. Since the problem is…
We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error must…
For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to…
Optimal power flow (OPF) is one of the most important optimization problems in the energy industry. In its simplest form, OPF attempts to find the optimal power that the generators within the grid have to produce to satisfy a given demand.…
Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…
Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not…
How to efficiently represent a graph in computer memory is a fundamental data structuring question. In the present paper, we address this question from a combinatorial point of view. A representation of an $n$-vertex graph $G$ is called…
The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the…
In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be…
In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way.…