The distinguishing numbers of graphs on closed surfaces

A graph GG is said to be dd-distinguishable   if there is a labeling c:V(G)→{1,2,…,d}c:V(G)→{1,2,…,d} such that no automorphism of GG other than the identity map preserves the labels of vertices given by cc. The smallest dd for which GG is dd-distinguishable is called the distinguishing number   of GG. We shall prove that every 4-representative triangulation on a closed surface, except the sphere, is 2-distinguishable after establishing a general theorem on the distinguishability of polyhedral graphs faithfully embedded on closed surfaces, and show that there is an upper bound for the distinguishing number of triangulations on a given closed surface, applying the re-embedding theory of triangulations.