For a Belyi function \(\beta :\mathbb{C}\mathbb{P}^1\rightarrow \mathbb{C}\mathbb{P}^1\) ramified only over the points \(-1\) , 1, and \(\infty \) , a corresponding “dessin d’enfant” \(\mathcal {D}_{\beta }\) is defined as the set \(\beta ^{-1}([-1,1])\) considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets \(\beta ^{-1}\{-1\}\) and \(\beta ^{-1}\{1\}\) , correspondingly. Merely the set \(\beta ^{-1}([-1,1])\) without a graph structure is called the support of \(\mathcal {D}_{\beta }\) . In this note, we solve the following problem: under what conditions different dessins \(\mathcal {D}_{\beta _1}\) and \(\mathcal {D}_{\beta _2}\) have equal supports?
