     # Adjacency Graphs of Polyhedral Surfaces

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in ℝ^3. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains K_5, K_5,81, or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K_4,4, and K_3,5 can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in Ω(n log n). From the non-realizability of K_5,81, we obtain that any realizable n-vertex graph has O(n^9/5) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

06/19/2020

### Universal Geometric Graphs

We introduce and study the problem of constructing geometric graphs that...
03/09/2020

### Adjacency Labelling for Planar Graphs (and Beyond)

We show that there exists an adjacency labelling scheme for planar graph...
07/11/2019

### The structure of k-planar graphs

Dujmović et al. (FOCS 2019) recently proved that every planar graph is a...
04/23/2018

### How to Realize a Graph on Random Points

We are given an integer d, a graph G=(V,E), and a uniformly random embed...
01/04/2021

### Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

We investigate special structures due to automorphisms in isogeny graphs...
03/16/2018

### Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

We study the problem of decomposing a volume bounded by a smooth surface...
09/08/2021

### Robust Numerical Integration on Curved Polyhedra Based on Folded Decompositions

We present a novel method to perform numerical integration over curved p...