Isomorphism in GRAPHS

Isomorphism of Graphs Definition: The simple graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic if there is a bijection (an one-to-one and onto function) f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f(a) and f(b) are adjacent in G2, for all a and b in V1. Such a function f is called an isomorphism. In other words, G1 and G2 are isomorphic if their vertices can be ordered in such a way that the adjacency matrices MG1 and MG2 are identical.

Graphs: Adjacency Matrix Example: A 1 2 3 4 1 a 2 d 4 b c 3 David Luebke 4 1/1/2019

Isomorphism of Graphs From a visual standpoint, G1 and G2 are isomorphic if they can be arranged in such a way that their displays are identical (of course without changing adjacency). Unfortunately, for two simple graphs, each with n vertices, there are n! possible isomorphisms that we have to check in order to show that these graphs are isomorphic. However, showing that two graphs are not isomorphic can be easy.

Isomorphism of Graphs For this purpose we can check invariants, that is, properties that two isomorphic simple graphs must both have. For example, they must have the same number of vertices, the same number of edges, and the same degrees of vertices. Note that two graphs that differ in any of these invariants are not isomorphic, but two graphs that match in all of them are not necessarily isomorphic.

Isomorphism of Graphs Example I: Are the following two graphs isomorphic? d a b c e d a b c e Solution: Yes, they are isomorphic, because they can be arranged to look identical. You can see this if in the right graph you move vertex b to the left of the edge {a, c}. Then the isomorphism f from the left to the right graph is: f(a) = e, f(b) = a, f(c) = b, f(d) = c, f(e) = d.

Isomorphism of Graphs Example II: How about these two graphs? d a b c Solution: No, they are not isomorphic, because they differ in the degrees of their vertices. Vertex d in right graph is of degree one, but there is no such vertex in the left graph.

Example of isomorphic graphs b 1 2 3 f 4 e 6 5 d c G An isomorphism between G and : a 6 d 5 b 1 e 2 c 3 f 4 Graph isomorphism is both very important and very intractable— Important: matching to known structures (eg chemical molecules, circuits, organizational solutions, visualizations) Intractable—no known good algorithms for determining if two graphs are isomorphic—brute force methods (eg start with one vertex, try all the ones it could match too, then try the next, etc) grow exponentially with the number of vertices. 1/25/2005 Tucker, Sec. 1.2

Isomorphic Graphs The two graphs below look different. Are from a graph theoretic point of view structurally the ‘same’? 2 b 1 3 a The renaming function f is an isomorphism. f.a = 1, f.b=3... Cycles and degree are preserved. c d 5 4 e

Graph Morphisms The two graphs below look different but are structurally the ‘same’. f.c 2(c) b f.a 1(a) 3(e) a c f.d The renaming function f is an isomorphism. f.a = 1, f.b=3... Cycles and degree are preserved. d 5(d) 4(b) e f.b f.e They are the ‘same’ up to the renaming of the vertices. These graphs are isomorphic.

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. 1/25/2005 Tucker, Sec. 1.2

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. 1/25/2005 Tucker, Sec. 1.2

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. Both edge reconstruction and vertex reconstruction are still open problems, The set of subgraphs you get by deleting each of the vertices (edges) singly is called the vertex (edge) deck of the graph. Can you prove that if two graphs have the same vertex/edge deck they must be isomorphic, or can you find a counter example? 1/25/2005 Tucker, Sec. 1.2

Two non-isomorphic graphs 1 4 a d 5 b e f g h 8 7 2 6 3 c 3 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! 1/25/2005 Tucker, Sec. 1.2

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. 1/25/2005 Tucker, Sec. 1.2

For the class to try: Are these pairs of graphs isomorphic? 3 Isomorphic: a-1, b-5, c-4, d-3, e-2, f-6. a e 5 1 #1 c d b f 2 6 4 d Not Isomorphic: 5 K3’s on left, 4 K3’s on right. 1 2 a b c 5 6 #2 7 3 e g 4 f 1/25/2005 Tucker, Sec. 1.2