  We define strongly chordal digraphs, which generalize strongly chordal graphs
and chordal bipartite graphs, and are included in the class of chordal
digraphs. They correspond to square 0,1 matrices that admit a simultaneous row
and column permutation avoiding the {\Gamma} matrix. In general, it is not
clear if these digraphs can be recognized in polynomial time, and we focus on
symmetric digraphs (i.e., graphs with possible loops), tournaments with
possible loops, and balanced digraphs. In each of these cases we give a
polynomial-time recognition algorithm and a forbidden induced subgraph
characterization. We also discuss an algorithm for minimum general dominating
set in strongly chordal graphs with possible loops, extending and unifying
similar algorithms for strongly chordal graphs and chordal bipartite graphs.
