Geometric graph theory

A geometric graph is a graph in which the vertices or edges are associated with geometric objects, the simplest realisation is a Random geometric graph.

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 the 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. The Hadwiger–Nelson problem concerns the chromatic number of these graphs.

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