Solution of Graph Isomorphism in current literature. As of 2008, the best algorithm for graph isomorphism (Babai & Luks 1983) has run time $2^{O(\sqrt(n log n))}$ for graphs with n vertices. Does this algorithm gives a yes / no answer or provide solution?
i.e.  , for $G,H$  graphs , does  this algorithm gives solution $P$ , so,  $PGP^{-1}=H$?
 A: Once you have an algorithm for deciding whether two graphs are isomorphic, you can use it to find the isomorphism whenever one exists.  Given isomorphic graphs $G$ and $H$, let $v$ be the first vertex of $G$ (meaning "first" in the ordering used to present the graph, for example as a matrix), and check, for each vertex $w$ of $H$ whether the induced subgraphs $G-\{v\}$ and $H-\{w\}$ are isomorphic via an isomorphism that preserves adjacency to $v$ and $w$. You'll find at least one $w$ for which the answer is "yes", and then you can start defining your isomorphism by sending $v$ to $w$.  Then go on to the next vertex of $G$ and find a vertex of $H$ that can serve as its image under the isomorphism.  Repeat this process through all the vertices of $G$, using your isomorphism-checking algorithm at all steps, until you've built the desired isomorphism.  This algorithm uses your isomorphism-checking algorithm $O(n^2)$ times, so the time complexity gets multiplied by $O(n^2)$, but that factor can be absorbed into the $O$ in the exponent of the time bound that you quoted. 
EDIT: When checking  whether $G-\{v\}$ is isomorphic to $H-\{w\}$, you need to make sure that the isomorphism sends the neighbors of $v$ in $G$ to exactly the neighbors of $w$ in $H$.  This seems to require an isomorphism test not for plain graphs $G-\{v\}$ and $H-\{w\}$ but for such graphs equipped with partition of the vertex sets into two pieces ("adjacent to the removed point" and "not adjacent").  Fortunately, isomorphism testing for such vertex-partitioned graphs can be reduced to isomorphism for ordinary graphs.  The idea is to enlarge the graphs $G-\{v\}$ and $H-\{w\}$ by adding some small subgraphs as "tags" to distinguish the two sorts of vertices.  
