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I am wondering how to prove that Subgraph Isomorphism is NP Complete. Wikipedia indicates that the CLIQUE problem can be used to demonstrate this, but I can't figure out how. I also found this link that demonstrates how Subgraph Isomorphism reduces to CLIQUE, but I can't figure out how to reverse it. Subgraph isomorphism reduction from the Clique problem

Here is a formal example of the problem from DASGUPTA 8.10: Given as input two undirected graphs G and H, determine whether G is a subgraph of H (that is, whether by deleting certain vertices and edges of H we obtain a graph that is, up to renaming of vertices, identical to G), and if so, return the corresponding mapping of V (G) into V (H).

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  • $\begingroup$ Well, what happens if you choose $G$ to be a clique with $k$ vertices? $\endgroup$ – dkaeae Mar 11 '19 at 16:18
  • $\begingroup$ Can you do that? If so, that is the answer because a clique will contain every possible subgraph of size k. But is it legal to constrain the input to a specific G that is fully connected? I thought it had to take any graphs G and H $\endgroup$ – DanGoodrick Mar 11 '19 at 16:20
  • $\begingroup$ The question you link actually shows that clique reduces to subgraph isomorphism, which is the exact thing you're looking for. $\endgroup$ – David Richerby Mar 11 '19 at 16:35
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    $\begingroup$ Possible duplicate of Subgraph isomorphism reduction from the Clique problem $\endgroup$ – David Richerby Mar 11 '19 at 16:35
  • $\begingroup$ The link I provided assumes a fully connected graph but my question only specifies any graph H. Is the reduction still possible/valid in this case? $\endgroup$ – DanGoodrick Mar 13 '19 at 12:01
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A subgraph is isomorphic to a clique if, and only if, it is a clique.

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  • $\begingroup$ But what if H doesn't have a clique the same size as G? There are lots of permutations of H that may or may not be isomorphic with G. $\endgroup$ – DanGoodrick Mar 11 '19 at 16:23
  • $\begingroup$ I'm not sure what you mean. If H is isomorphic to G, any permutation of H is also isomorphic. $\endgroup$ – David Richerby Mar 11 '19 at 16:36
  • $\begingroup$ I thought H could be any graph, not just a graph that is isomorphic to G. $\endgroup$ – DanGoodrick Mar 13 '19 at 11:58
  • $\begingroup$ We're trying to reduce $k$-clique to sub graph isomorphism. That means we're given a graph $F$ and a number $k$, and we need to produce graphs $G$ and $H$ such that $G$ has a subgraph isomorphic to $H$ if, and only if, $F$ has a clique of size at least $k$. We get to choose the input to subgraph isomorphism. $\endgroup$ – David Richerby Mar 13 '19 at 12:42
  • $\begingroup$ Thanks for addressing my concern. So in my other reduction exercises, I have taken the input to a known NP Complete problem, modified it, and produced the same answer as the known NPC but with a unproven NP Complete problem. For example, I took the input for SAT (known NPC) and modified it to be a valid input for 3SAT. Then I modified the output of 3SAT to give the same result as would have been obtained by SAT. So, that is what I know how to do. This seems to be a different problem. A cascade of sorts. Am I correct in my understanding? $\endgroup$ – DanGoodrick Mar 13 '19 at 15:34

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