Graph Theory Planarity and Eulers Formula Definitions Crossing

Graph Theory Planarity and Euler’s Formula

Definitions • Crossing: a place in a graph where it looks like two edges intersect but not at their endpoints • Planar graph: a graph G is planar if it can be drawn in such a way that the edges only intersect at the vertices, i. e. no crossings • Planar representation: a drawing of a planar graph G in which edges only intersect at vertices

Example 1 • Draw of planar representation of the following graph, if possible.

Example 2 • Are the following graphs planar?

How can we tell if the graph is planar? • Region: a maximal section of the plane in which any two points can be joined by a curve that does not intersect any part of G. • Or an area bounded by edges

How can we tell if the graph is planar? • Study these planar graphs and see if you can find a pattern. (n = vertices, q = edges and r = regions)

Euler’s Formula • If G is a connected planar graph with n vertices, q edges and r regions, then n – q + r = 2. • Another cool property: if G is a planar graph with n greater than or equal to 3 vertices and q edges, then. Furthermore, if equality holds, then every region is bounded by three edges.

Graphs that are Nonplanar • There are two common graphs that are nonplanar. • Also any graph that has these graphs as part of them are nonplanar.

Regular Polyhedra • Polyhedron: a solid bounded by flat surfaces • If we “deflate” a polyhedron we can create a graph

Regular Polyhedra • The neat thing about any polyhedra is that all their graphs are planar • They follow Euler’s Formula but this time we use vertices, edges and faces (instead of regions) • So if a polyhedron has V vertices, E edges, and F faces, then V – E + F = 2.

How many regular polyhedra exist?