Forests and trees among Gallai graphs

The Gallai graph Gal(G) of a graph GG has the edges of GG as its vertices, and two distinct vertices ee and ff of Gal(G) are adjacent in Gal(G) if the edges ee and ff of GG are adjacent in GG but do not span a triangle in GG. In the present paper we characterize those graphs whose Gallai graphs are forests or trees, respectively.