  This thesis provides an explanation of L\'aszl\'o Babai's quasi-polynomial
algorithm for the Graph Isomorphism Problem published in 2015 with a particular
focus on the case of local certificates, i.e. the case that cannot be dealt
with by Luks' method. The thesis extends the explanations provided by Harald
Andr\'es Helfgott in 2017. It is concluded that the complexity of Babai's
algorithm is $\exp\left(C \left(\log n\right)^3\right)$ for $n$ the number of
vertices, $C$ a constant. Group theoretical and combinatorial arguments are
used to give more details on Babai's method of local certificates. They treat
Luks' barrier case in which the imprimitve permutation group $G$ can be mapped
onto an alternating group with large domain.
