MAT 2720 Discrete Mathematics Section 8.7 Planar Graphs http://myhome.spu.edu/lauw

Minimum to copy for this section... Proofs with graphical components

Goals Define Planar Graphs The conditions for a graph to be planar Series Reductions Homeomorphic Graphs

Example 1 The following are 2 ways of drawing the same graph, K4.

Definition A graph is planar if it can be drawn in the plane without its edges crossing.

Definition A graph is planar if it can be drawn in the plane without its edges crossing. K4 is planar K5 is NOT planar K3,3 is NOT planar

Definition A graph is planar if it can be drawn in the plane without its edges crossing. K4 is planar K5 is NOT planar K3,3 is NOT planar We will look at why.

Faces of a Planar Graph

Euler’s Formula for Graphs If G is a connected, planar graph with e edges, v vertices, and f faces, then f=e-v+2

Euler’s Formula for Graphs If G is a connected, planar graph with e edges, v vertices, and f faces, then f=e-v+2

Example 2 K3,3 is NOT planar

Example 2: Proof by Contradiction Suppose K3,3 is planar 1. Every cycle has at least 4 edges.

Example 2: Proof by Contradiction Suppose K3,3 is planar 1. Every cycle has at least 4 edges. 2.The no. of edges that bound faces is at least 4f (with some edges counted twice). f=e-v+2

Observations A graph contains K3,3 or K5 as a subgraph is NOT planar.

Eliminating edges (𝑎, 𝑏), (𝑏, 𝑐), and (𝑐, 𝑎) Formal Solutions Eliminating edges (𝑎, 𝑏), (𝑏, 𝑐), and (𝑐, 𝑎) Since the graph contains a subgraph of 𝐾3,3, it is not planar.

Observations A graph contains a graph “somewhat” similar to K3,3 or K5 as a subgraph is NOT planar.

Series Reduction Edges in Series Series Reduction

Homeomorphic Two graphs are homeomorphic if they can be reduced to isomorphic graphs by a sequence of series reduction.

Example 3 Show that the following graphs are homeomorphic.

Formal Solutions The graphs are homeomorphic since they can be reduced to the same graph by a sequence of series reduction. Series Reduction: eliminating vertices 𝑎 and 𝑏 Series Reduction: eliminating vertices 𝑐 and 𝑑

Back to our Earlier Example...

Formal Solutions Series Reduction: eliminating vertex 𝑑 Eliminating edges (𝑎, 𝑏), (𝑏, 𝑐), and (𝑐, 𝑎) Since the graph contains a subgraph homeomorphic to 𝐾3,3, it is not planar

Kuratowski’s Theorem A graph is planar iff it does not contain a subgraph homeomorphic to 𝐾3,3 or 𝐾5 .

Example 3 Show that the following graph is not planar.

Example 3 Key: Locate the subgraph homeomorphic to K3,3 or K5

Example 3: Formal Solutions Eliminating edges (a,b), (f,e), and (g,h) eliminating vertices g and h

Example 3: Formal Solutions Eliminating edges (a,b), (f,e), and (g,h) eliminating vertices g and h Since the graph contains a subgraph homeomorphic to 𝐾3,3, it is not planar