Let R be a compact, connected, orientable surface of genus g with n boundary components. We prove that edge preserving maps of the curve graphs on R are induced by homeomorphisms of R when g=0, n >= 5 or g = 1, n >= 3 or g >= 2. We will also talk about similar results for the edge preserving maps of the nonseparating curve graphs, and the rectangle preserving maps of the Hatcher-Thurston graphs. Speaker(s): Elmas Irmak (Bowling Green State University / UM)