Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Geometric graph theory
Summary
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices; thus, it can be described as "the theory of geometric and topological graphs" (Pach 2013). Geometric graphs are also known as spatial networks.
A planar straight-line graph is a graph in which the vertices are embedded as points in the Euclidean plane, and the edges are embedded as non-crossing line segments. Fáry's theorem states that any planar graph may be represented as a planar straight line graph. A triangulation is a planar straight line graph to which no more edges may be added, so called because every face is necessarily a triangle; a special case of this is the Delaunay triangulation, a graph defined from a set of points in the plane by connecting two points with an edge whenever there exists a circle containing only those two points.
The 1-skeleton of a polyhedron or polytope is the set of vertices and edges of said polyhedron or polytope. The skeleton of any convex polyhedron is a planar graph, and the skeleton of any k-dimensional convex polytope is a k-connected graph. Conversely, Steinitz's theorem states that any 3-connected planar graph is the skeleton of a convex polyhedron; for this reason, this class of graphs is also known as the polyhedral graphs.
A Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points. The Euclidean minimum spanning tree is the minimum spanning tree of a Euclidean complete graph. It is also possible to define graphs by conditions on the distances; in particular, a unit distance graph is formed by connecting pairs of points that are a unit distance apart in the plane.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.