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Figure 15.7 Isomorphic graphs
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because there are n! different possibilities. For example, there are 8! = 40,320 different possible ways to assign the 8 labels to the 8 vertices of each graph in Example 15.8. The following algorithm is more efficient: 1. Arbitrarily label the vertices of one graph. (Assume here that the positive integers are used for labels.) 2. Find a vertex on the second graph that has the same degree as vertex 1 on the first graph, and number that vertex 1 also. 3. Label the vertices that are adjacent to the new vertex 1 with the same numbers that correspond to the vertices that are adjacent to the other vertex 1. 4. Repeat step 3 for each of the other newly labeled vertices. If at some point in the process, step 3 is not possible, then backtrack and try a different labeling. If no amount of backtracking seems to help, try proving that the two graphs are not isomorphic. To prove that two graphs are not isomorphic (by definition) would require showing that every one of the possible n! different labellings fails to preserve adjacency. That is impractical. The following theorem makes it much easier to prove that two graphs are not isomorphic. Theorem 15.4 Isomorphism Tests for Graphs All of the following conditions are necessary for two graphs to be isomorphic: 1. They must have the same number of vertices. 2. They must have the same number of edges. 3. They must have the same number of connected components. 4. They must have the same number of vertices of each degree. 5. They must have the same number of cycles of each length. EXAMPLE 15.9 Proving that Two Graphs Are Not Isomorphic
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Figure 15.8 shows three graphs, to be compared with the two isomorphic graphs in Figure 15.7. Each of these graphs has eight vertices, so each could be isomorphic to those two graphs. Graph G1 is not isomorphic to those two graphs because it has only 14 edges. The graphs in Figure 15.7 each have 16 edges. Condition 2 of Theorem 15.4 says that isomorphic graphs must have the same number of edges. Graph G2 does have 16 edges. But it is not isomorphic to the two graphs in Figure 15.7 because it has two connected components. Each of the two graphs in Figure 15.7 has only one connected component.
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Figure 15.8 Possibly isomorphic graphs
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Condition 3 of Theorem 15.5 says that isomorphic graphs must have the same number of connected components. Graph G3 has 16 edges and only one connected component. But it is still not isomorphic to the two graphs in Figure 15.7 because it has some vertices of degree 3 (and some of degree 5). All the vertices of the two graphs in Figure 15.7 have degree 4. Condition 4 of Theorem 15.5 says that isomorphic graphs must have the same number of vertices of each degree.
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Note that in Example 15.9 we really only have to compare each graph with one of the two graphs in Figure 15.7 on page 289, not both of them. Theorem 15.5 Graph Isomorphism Is an Equivalence Relation The isomorphism relation among graphs satisfies the three properties of an equivalence relation: 1. Every graph is isomorphic to itself. 2. If G 1 is isomorphic to G 2 then G 2 is isomorphic to G 1. 3. If G 1 is isomorphic to G 2 and G 2 is isomorphic to G 3 , then G 1 is isomorphic to G 3. THE ADJACENCY MATRIX FOR A GRAPH An adjacency matrix for a graph (V, E) is a twodimensional boolean array
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obtained by ordering the vertices V = {v0 , v1, ..., vn 1} and then assigning true to a[i][j] if and only if vertex vi is adjacent to vertex vj. EXAMPLE 15.10 An Adjacency Matrix