Isomorphism

1
Section 7.3

• Isomorphism
• and planarity

2
Isomorphism

• One of these things is not like the other

3
Isomorphism

• Graphs G and H are isomorphic if one can be
  redrawn to look like the other.
• Graphs G and H are not isomorphic if
• they do not have the same number of nodes or
  edges
• they do not have the same degree sequence
• they do not have the same number of cycles of
  any particular size
• etc.

4
Example

• Explain why the two graphs below are not
  isomorphic.

5
Practice

• Find all isomorphic pairs among the graphs below.

6
Planar Graphs

• Given three houses, is there a way to connect
  them to their electricity, cable TV, and
  telephone companies without one services wires
  crossing over the other?

7
Definitions

• A simple, connected graph is called planar if
  there is a way to draw it (on a plane) so that no
  edges cross.
• A graph is called bipartite if its set of nodes
  can be partitioned into two disjoint sets S and T
  so that every edge in the graph has one endpoint
  in S and one in T.
• A complete graph on n nodes, Kn, is the simple
  graph with nodes 1,2,,n and an edge between
  every pair of distinct nodes.
• A complete bipartite graph on n,m nodes, denoted
  Kn,m is the simple bipartite graph with edges
  connecting each node of the n nodes in S to each
  of the m nodes in T.

8
Is the graph K4 planar?
9
Is the graph K3,2 planar?
10
Non-Planar Graphs

• Note K5 and K3,3 are non-planar graphs.
• Utilities Problem