Planarity and Euler’s Formula 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?