ICS 241

1
ICS 241

• Discrete Mathematics II
• William Albritton, Information and Computer
  Sciences Department at University of Hawaii at
  Manoa
• For use with Kenneth H. Rosens Discrete
  Mathematics Its Applications (5th Edition)
• Based on slides originally created by
• Dr. Michael P. Frank, Department of Computer
  Information Science Engineering at University
  of Florida

2
Section 8.3 Graph Representations Isomorphism

• Graph representations
• Adjacency lists.
• Adjacency matrices.
• Incidence matrices.
• Graph isomorphism
• Two graphs are isomorphic if they are identical
  except for their node names.

3
Adjacency Lists

• A table with 1 row per vertex, listing its
  adjacent vertices.

b
a
d
c
e
f
4
Adjacency Matrices

• A way to represent simple graphs
• Matrix Aaij, where aij is 1 if vi, vj is an
  edge of G, and is 0 otherwise.
• Can extend to multigraphs, pseudographs, directed
  graphs, and directed multigraphs by letting each
  matrix elements be the number of links (possibly
  gt1) between the nodes.

5
Incidence Matrices

• Another way to represent simple graphs,
  multigraphs, or pseudographs
• Matrix Mmij, where mij is 1 if edge ej is
  incident with vertex vi, and is 0 otherwise.
• Rows represent vertices (vi) and columns
  represent edges (ej)

6
Class Exercise

• Exercise 3, 7 (p. 563)
• Represent the graph in 3 as an adjacency list
  and as an adjacency matrix

7
Graph Isomorphism

• Formal definition
• Simple graphs G1(V1, E1) and G2(V2, E2) are
  isomorphic if ? a bijection fV1?V2 such that ?
  a,b?V1, a and b are adjacent in G1 if f(a) and
  f(b) are adjacent in G2.
• f is the renaming function between the two node
  sets that makes the two graphs identical.
• This definition can easily be extended to other
  types of graphs.

8
Graph Invariants under Isomorphism

• Necessary but not sufficient conditions for
  G1(V1, E1) to be isomorphic to G2(V2, E2)
• We must have that V1V2, and E1E2.
• The number of vertices with degree n is the same
  in both graphs.
• For every proper subgraph g of one graph, there
  is a proper subgraph of the other graph that is
  isomorphic to g.

9
Isomorphism Example

• If isomorphic, label the 2nd graph to show the
  isomorphism, else identify difference.

d
b
b
a
a
d
c
e
f
e
c
f
10
Are These Isomorphic?

• If isomorphic, label the 2nd graph to show the
  isomorphism, else identify difference.

• Same of vertices

a
b

• Same of edges

• Different of verts of degree 2! (1 vs 3)

d
e
c
11
Class Exercise

• Exercise 43, 57.c., 63. (p. 565)
• Each pair of students should use only one sheet
  of paper while solving the class exercises