  The Gallai graph $\Gamma(G)$ of a graph $G$ has the edges of $G$ as its
vertices and two distinct vertices $e$ and $f$ of $\Gamma(G)$ are adjacent in
$\Gamma(G)$ if the edges $e$ and $f$ of $G$ are adjacent in $G$ but do not span
a triangle in $G$. Clearly, $\Gamma(G)$ is a subgraph of the line graph of $G$.
While line graphs can be recognized efficiently the complexity of recognizing
Gallai graphs is unknown. In the present paper we characterize those graphs
whose Gallai graphs are forests or trees, respectively.
