Planar Graph Theory We say that a graph is planar if it can be drawn in the plane without edges crossing. We use the term plane graph to refer to a planar depiction of a planar graph.
e.g. K4 is a planar graph
Q1: The following is also planar. Find a plane graph version of the graph.
A B
F
E
D C
A Method that sometimes works for drawing the plane graph for a planar graph: 1. Find the largest cycle in the graph. 2. The remaining edges must be drawn inside/outside the cycle so that they do not cross each other.
Q2: Using the method above, find a plane graph version of the graph below.
A B C D
E F G H
non e.g. K3,3: K5
Here are three (plane graph) depictions of the same planar graph:
J N M J K J N I M K K I N M I O O L O
L L
A face of a plane graph is a region enclosed by the edges of the graph. There is also an unbounded face, which is the outside of the graph.
Q3: For each of the plane graphs we have drawn, find: V = # of vertices of the graph E = # of edges of the graph F = # of faces of the graph
Q4: Do you have a conjecture for an equation relating V, E and F for any plane graph G?
Q5: Can you name the 5 Platonic Solids (i.e. regular polyhedra)? (This is a geometry question.)
Q6: Find the # of vertices, # of edges and # of faces for each Platonic Solid.
Q7: Find an equation relating V, E and F for the 5 Platonic Solids. Connected Graphs and Trees
Formal Definition: We say a graph G is connected if there is a path between every pair of vertices.
e.g. EVERY graph we have done so far.
Non e.g. The following graph G, where V = {A,B,C,D,E,F,U,W} B E
C U F
W A D
A graph G is called a tree if it is (1) connected and (2) acyclic (i.e. has no cycles). e.g. Graph G below:
I R P S T J U O K M Q
L N
Q8: Draw a tree on 6 vertices, one on 8 vertices and one on 9 vertices.
Q9: Can you draw a tree without at least two vertices of degree 1?
Q10: In a tree T, take any two vertices u,v in T. How many paths are there between u and v?
Q11: How many edges are there in a tree on n vertices? Q12: The cast of Lost has decided to stay on the island. They have five cities on the island. Below is a graph showing the cost (in coconuts) to build a road between each city. If there is not an edge between two cities, then there is a mountain in the way and the road cannot be built. Determine the least cost of making all the cities reachable from each other.
NorthTown
9 3 5 11 8 Palm Leaf New Easy 9
7 3
10
Plane City Cabo San Lost
Def’n: A spanning tree of a connected graph G is a connected acyclic subgraph T of G. e.g. Above, you found a spanning tree of the island of Lost.
Note: A graph can have its edges weighted, as above. We call this a weighted graph.
Def’n: A minimum weight spanning tree is (of course) a spanning tree of min. wt.
Q13: Find a minimum weight spanning tree of the following graph G:
15
13 13 16
14 14 12 12
16 14 14 15