Mapping graph to another graph's sub-graph
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 $k$&nbsp;vertices to solve CLIQUE on&nbsp;$G'$. Unless you can restrict the form of $G$ and/or&nbsp;$G'$, 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&nbsp;$G'$ 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 $\epsilon&gt;0$. (The algorithm was originally announced as having quasipolynomial running time but an error in the analysis came to light in January 2017.)