Planar Graph

Description:

• A graph that can be drawn in the plane without any edges crossing
  • where a crossing of edges is the intersection of the lines or arcs representing them at a point other than their common endpoint.
  • such drawing is called “Planar representation”
  • 
  • theorem Any subgraph of a planar graph is also a planar
  • theorem Merging two adjacent vertices of a planar graph leaves another planar graph
  • Tree is also planar graph
• A planar can be splits into regions
  • 
  • each region is the space between edges
  • also count the outer region

Euler’s formula:

• Let $G$ be a connected planar simple graph with $e$ edges and $v$ vertices.
• Let $r$ be the number of regions in a planar representation of $G$
• Then $r=e-v+2$
• Corollary 1:
  • theorem If $G$ is a connected planar simple graph, with $e$ edges and $v$ vertices, where $v\ge 3$
  • then $e\le 3v-6$
• Corollary 2:
  • theorem If 𝐺 is a connected planar simple graph
  • then 𝐺 has a vertex of $deg\le 5$
• Corollary 3:
  • theorem If a connected planar simple graph has 𝑒 edges and 𝑣 vertices with 𝑣 ≥ 3 and no circuits of length three
  • then $e\le 2v-4$

Homeomorphic:

• theorem The graph $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ are called homeomorphic if they can be obtained from the same graph by a sequence of elementary subdivisions
  • Elementary Subdivision is when remove an edge $\{u,v\}$ and add new vertex $w$ together with edge $\{u,w\}$ and $\{w,v\}$
  • 

Non planar

• theorem A graph is non-planar IFF it contains a subgraph Homeomorphic to $K_5$ or $K_{3,3}$
  •