Isomorphism Testing Examples

Graph Theory · Axiom Academy

EXAMPLE Isomorphism Testing Examples Master the art of determining whether two graphs are isomorphic Excellent work! You've mastered the essential techniques for testing graph isomorphism. Here's what we learned: Start with invariants: Always check basic properties first (number of vertices, edges, degree sequences). These can quickly prove non-isomorphism. Structural properties matter: Look for triangles, cycles, connectivity patterns, and other structural features that must be preserved under isomorphism. Finding bijections: For small graphs, systematically try to construct a vertex mapping that preserves adjacency. Start with vertices of unique degrees. Recognize common graphs: Familiarity with standard graph families (paths, cycles, complete graphs, complete bipartite graphs) helps identify disguised versions quickly. Verification is crucial: Once you propose a bijection, verify that it preserves all edge relationships in both directions. These techniques form the foundation for analyzing graph structure. Practice with various graph types to develop your intuition!

This is the written version of the interactive lesson above. See the full Graph Theory course.