1. 1/25/2005Tucker, Sec. 1.21 Applied Combinatorics, 4th Ed. Alan Tucker Section 1.2 Isomorphism Prepared by Jo Ellis-Monaghan

2. 1/25/2005Tucker, Sec. 1.22 Two graphs G and are isomorphic if : –There exists a one-to-one correspondence between vertices in G and, such that –There is an edge between a and b in G if and only if there is an edge between the corresponding vertices and in. The definition for oriented graphs is the same, except the head and tail of each edge of G must correspond to the head and tail in. Definition of Isomorphism

3. 1/25/2005Tucker, Sec. 1.23 Example of isomorphic graphs ab cd e f 12 3 4 56 G An isomorphism between G and : a 6d 5 b1e2 c3f4

4. 1/25/2005Tucker, Sec. 1.24 K n, the complete graph on n vertices K2K2 K3K3 K5K5 K8K8 K6K6 K4K4 K1K1

5. 1/25/2005Tucker, Sec. 1.25 The complement of a graph The complement of G has all the edges that are missing in G— i.e. that would have to be added to make the complete graph. G K6K6

6. 1/25/2005Tucker, Sec. 1.26 Advantage of the complement Theorem: Two graphs, G and H, are isomorphic if and only if their complements are. In practice this means that we work with whichever of G or has few edges.

7. 1/25/2005Tucker, Sec. 1.27 Subgraphs Definition: if G is a graph, a subgraph H of G consists of a subset V of the vertices of G, and a subset of the edges of G with endpoints in V. n a b c d e f g h i j k l m o A graph G cde i jk h g m f l Two subgraphs of G

8. 1/25/2005Tucker, Sec. 1.28 Induced subgraphs Choose a subset of the vertices of G, then include only the edges among those vertices. n a b c d e f g h i j k l m o A graph G Subgraph induced by the vertices of degree 4. d e h j k d e h j k

9. 1/25/2005Tucker, Sec. 1.29 Elementary properties of isomorphic graphs Edge and vertex counts –Isomorphic graphs have the same number of edges and vertices. Vertex sequence (the list of vertex degrees) –Isomorphic graphs have the same vertex sequences. Warning!! These can be used to show two graphs are not isomorphic, but can not show that two graphs are isomorphic.

10. 1/25/2005Tucker, Sec. 1.210 Two non-isomorphic graphs Vertices: 6 Edges: 7 Vertex sequence: 4, 3, 3, 2, 2, 0. Vertices: 6 Edges: 7 Vertex sequence: 5, 3, 2, 2, 1, 1.

11. 1/25/2005Tucker, Sec. 1.211 Subgraph properties of isomorphic graphs Isomorphic graphs have the same sets of subgraphs: – there is a one-to-one correspondence between the subgraphs such that corresponding subgraphs are isomorphic. Typically check induced subgraphs, or number of a specific kind of subgraphs such as cycles or cliques. Warning!! These can be used to show two graphs are not isomorphic, but can not show that two graphs are isomorphic.

12. 1/25/2005Tucker, Sec. 1.212 Two non-isomorphic graphs a c d b e f g h 23 5 6 7 8 14 Vertices: 8 Edges: 10 Vertex sequence: 3, 3, 3, 3, 2, 2, 2, 2. Vertices: 8 Edges: 10 Vertex sequence: 3, 3, 3, 3, 2, 2, 2, 2. However, induced subgraphs on degree 3 vertices are NOT isomorphic! 3

13. 1/25/2005Tucker, Sec. 1.213 An approach to checking isomorphism: Count the vertices. The graphs must have an equal number. Count the edges. The graphs must have an equal number. Check vertex degree sequence. Each graph must have the same degree sequence. Check induced subgraphs for isomorphism. If the subgraphs are not isomorphic, then the larger graphs are not either. Count numbers of cycles/cliques. If these tests don’t help, and you suspect the graphs actually are isomorphic, then try to find a one-to-one correspondence between vertices of one graph and vertices of the other. Remember that a vertex of degree n in the one graph must correspond to a vertex of degree n in the other.

14. 1/25/2005Tucker, Sec. 1.214 For the class to try: a b f e cd 1 2 3 4 5 6 Are these pairs of graphs isomorphic? #1 #2 Isomorphic: a-1, b-5, c-4, d-3, e-2, f-6. Not Isomorphic: 5 K 3 ’s on left, 4 K 3 ’s on right. abc d f ge 1 2 3 4 56 7