Skeletonization via Local Separators

We propose a new algorithm for curve skeleton computation that differs from previous algorithms by being based on the notion of local separators . The main benefits of this approach are that it is able to capture relatively fine details and that it works robustly on a range of shape representations. Specifically, our method works on shape representations that can be construed as spatially embedded graphs. Such representations include meshes, volumetric shapes, and graphs computed from point clouds. We describe a simple pipeline where geometric data are initially converted to a graph, optionally simplified, local separators are computed and selected, and finally a skeleton is constructed. We test our pipeline on polygonal meshes, volumetric shapes, and point clouds. Finally, we compare our results to other methods for skeletonization according to performance and quality.

Eulerian and Even-Face Graph Partial Duals

Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial duals of cellularly embedded graphs, and characterize Eulerian and even-face graph partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph.

Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus g ≥ 0 of the graph and the number of terminals t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value G of the genus.

Facial Homogeneous Colouring of Graphs

A proper colouring of a plane graph G is called facially homogeneous if it uses the same number of colours for every face of G. We study various sufficient conditions of facial homogeneous colourability of plane graphs, its relation to other facial colourings, and the extension of this concept for embedded graphs in general.

Reconstructing linearly embedded graphs: A first step to stratified space learning

<p style='text-indent:20px;'>In this paper, we consider the simplest class of stratified spaces – linearly embedded graphs. We present an algorithm that learns the abstract structure of an embedded graph and models the specific embedding from a point cloud sampled from it. We use tools and inspiration from computational geometry, algebraic topology, and topological data analysis and prove the correctness of the identified abstract structure under assumptions on the embedding. The algorithm is implemented in the Julia package Skyler, which we used for the numerical simulations in this paper.</p>