Conjugation of Transitive Permutation Pairs and Dessins d'Enfants

Let E be a finite set. Given permutations x and y of E that together generate a transitive subgroup, for which s is it true that x and the conjugate of y by s also generate a transitive subgroup? Such transitive permutation pairs encode dessins d'enfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group acts on dessins d'enfants and permutes them in a very mysterious way. Two dessins d'enfants that share certain elementary combinatorial features are related by conjugations as above, and dessins d'enfants in the same Galois-orbit share these features and more, so it seems worthwhile to have a good answer to the above question. I classify, relative to x and y, exactly those transpositions s for which the new pair is guaranteed to be transitive. I also provide examples of the "exceptional" s which show the range of possible behavior and prove that the above question for the exceptional cases is equivalent to a natural question about deletion in graphs that may have a good answer in this more structured world of topological graphs. Finally, I classify transpositions s according to how they change the genus of the surface underlying the dessin d'enfant of x, y. Some of the tools, like the Reroute Operation/Theorem, may have use beyond Dessins d'Enfants.