Applied Combinatorics, 4th Ed. Alan Tucker

Applied Combinatorics, 4th Ed.Alan Tucker Section 1.2 Isomorphism Prepared by Jo Ellis-Monaghan Tucker, Sec. 1.2

Definition of Isomorphism • 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 . Tucker, Sec. 1.2

An isomorphism between G and : a 6 d 5 b 1 e 2 c 3 f 4 Example of isomorphic graphs a b 1 2 3 f 4 e 6 5 d c G Tucker, Sec. 1.2

K6 Kn, the complete graph on n vertices K1 K4 K3 K2 K8 K5 Tucker, Sec. 1.2

K6 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 Tucker, Sec. 1.2

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. Tucker, Sec. 1.2

g a b g f c e d m h j k i l m h o n A graph G Subgraphs • Definition: if G is a graph, a subgraphH of G consists of a subset V of the vertices of G, and a subset of the edges of G with endpoints in V. c d e f i l j k Two subgraphs of G Tucker, Sec. 1.2

e e d d h h j j k k Induced subgraphs • Choose a subset of the vertices of G, then include only the edges among those vertices. a b g f c e d h j k i l m o n Subgraph induced by the vertices of degree 4. A graph G Tucker, Sec. 1.2

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. 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. Tucker, Sec. 1.2

Subgraph properties ofisomorphic 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. Tucker, Sec. 1.2

e h f g b Two non-isomorphic graphs 1 4 d a 5 8 7 6 2 3 3 c 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! 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. 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. 5 e a 1 #1 c d b f 6 2 4 d Not Isomorphic: 5 K3’s on left, 4 K3’s on right. 2 1 a b c 5 6 #2 7 3 4 e g f Tucker, Sec. 1.2

Applied Combinatorics, 4th Ed. Alan Tucker