Almost-regular Dessins d'Enfant on a torus and sphere

The Hurwitz problem asks which ramification data are realizable, i.e., appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.