1. Graph pattern matching Graph GGraph Qisomorphism f(a) = 1 f(b) = 6 f(c) = 8 f(d) = 3 f(g) = 5 f(h) = 2 f(i) = 4 f(j) = 7

2. 1/25/2005Tucker, Sec. 1.22 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.

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 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.

5. 1/25/2005Tucker, Sec. 1.25 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

6. More Example of Simple and Dual Simulation 100 200 A B 1 2 B B 3 4 B B 5 6 B D 7 K 8 A A0, 8, 9 B1, 2, 3, 4, 5 9 A Simple Simulation A0, 8, 9 B1, 2, 3, 4, 5 Dual Simulation 0 1 A B 4 B 8 A 1 2 B B 3 4 B 5 8 A B B 0 A 0 A 3 B Q G

7. More Example of Simple and Dual Simulation 10 30 A C A1 B2 C3 D4, 5 E7, 8, 11 F6, 9, 10 Simple Simulation Q G 40 20 6050 D B E F 1 2 A B 4 3 6 D C F 5 D E 11 10 98 F F E E 7

8.  Strong simulation: Define strong simulation by enforcing two conditions on simulation : duality and locality. Balls. For a node v in a graph G and a non-negative integer r, the ball with center v and radius r is a subgraph of G, denoted by ˆG[v, r], such that 1. for all nodes v in ˆG[v, r], the shortest distance dist(v, v) ≤ r, 2. it has exactly the edges that appear in G over the same node set. denoted by Q ≺ D L G, if there exist a node v in G and a connected subgraph Gs of G such that 1. Q ≺ D Gs, with the maximum match relation S; 2. Gs is exactly the match graph w.r.t. S 3. Gs is contained in the ball ˆG[v, dQ], where dQ is the diameter of Q. Graph Simulation 100 200 4 321 321 Q P P PP P P P P P P1, 2, 3, 4 4 3 1 P P P G 4 21 P P 4 32 P P P

9. More Example of Strong Simulation 100 200 A B 1 2 B A 3 4 B A 5 6 B B 0 A Q G 1 6 B 0 B A 1 2 B A 0 A 1 2 B A 3 B 2 A 3 4 B A 3 4 B A 5 B 4 A 5 6 B A 5 6 B B 0 A