How to solve the induced sub-graph isomorphism problem?
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This is the induced subgraph isomorphism problem, which was one of the problems proven NP-complete in Cook's original paper on NP-completeness. For a quick way to see that it's NP-complete, you can let $G$ be a complete graph on $k$ vertices to solve CLIQUE on $G'$. Unless you can restrict the form of $G$ and/or $G'$, you're probably out of luck in your quest for a good algorithm.
The problem seems to be easier if you insist that $G$ and $G'$ have the same number of vertices. In this case, you have the graph isomorphism problem. This is generally thought not to be NP-complete and, in December 2015, Babai announced an subexponential-time algorithm, which runs in time $\exp\exp(\tilde{O}(\sqrt{\log n}))$, where $\tilde{O}$ denotes hidden factors of $O(\log\log n)$. This is faster than $O(\mathrm{e}^{n^\epsilon})$ for all $\epsilon>0$. (The algorithm was originally announced as having quasipolynomial running time but an error in the analysis came to light in January 2017.)
David Richerby
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To answer the updated question, yes, it can be solved with a SAT solver. The problem is NP-complete, so it is in NP, so therefore it can be reduced to SAT, i.e., expressed in CNF-SAT (see What is the definition of P, NP, NP-complete and NP-hard?). Then you can use an off-the-shelf SAT solver to solve here.
For general techniques, take a look at Converting (math) problems to SAT instances and NP to SAT. How does it works?.
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