Decision Problem and Computational Complexity
To prove subgraph isomorphism NP-complete, it must be formulated as a decision problem. The input to the decision problem is a pair of graphs G and H. The answer to the problem is positive if H is isomorphic to a subgraph of G, and negative otherwise.
The proof of subgraph isomorphism being NP-complete is simple and based on reduction of the clique problem, an NP-complete decision problem in which the input is a single graph G and a number k, and the question is whether G contains a complete subgraph with k vertices. To translate this to a subgraph isomorphism problem, simply let H be the complete graph Kk; then the answer to the subgraph isomorphism problem for G and H is equal to the answer to the clique problem for G and k. Since the clique problem is NP-complete, this polynomial-time many-one reduction shows that subgraph isomorphism is also NP-complete.
An alternative reduction from the Hamiltonian cycle problem translates a graph G which is to be tested for Hamiltonicity into the pair of graphs G and H, where H is a cycle having the same number of vertices as G. Because the Hamiltonian cycle problem is NP-complete even for planar graphs, this shows that subgraph isomorphism remains NP-complete even in the planar case.
Subgraph isomorphism is a generalization of the graph isomorphism problem, which asks whether G is isomorphic to H: the answer to the graph isomorphism problem is true if and only if G and H both have the same number of vertices and the subgraph isomorphism problem for G and H is true. However the complexity-theoretic status of graph isomorphism remains an open question.
In the context of the Aanderaa–Karp–Rosenberg conjecture on the query complexity of monotone graph properties, Gröger (1992) showed that any subgraph isomorphism problem has query complexity Ω(n3/2); that is, solving the subgraph isomorphism requires an algorithm to check the presence or absence in the input of Ω(n3/2) different edges in the graph.
Read more about this topic: Subgraph Isomorphism Problem
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