  We study the class of 1-perfectly orientable graphs, that is, graphs having
an orientation in which every out-neighborhood induces a tournament.
1-perfectly orientable graphs form a common generalization of chordal graphs
and circular arc graphs. Even though they can be recognized in polynomial time,
little is known about their structure. In this paper, we develop several
results on 1-perfectly orientable graphs. In particular, we: (i) give a
characterization of 1-perfectly orientable graphs in terms of edge clique
covers, (ii) identify several graph transformations preserving the class of
1-perfectly orientable graphs, (iii) exhibit an infinite family of minimal
forbidden induced minors for the class of 1-perfectly orientable graphs, and
(iv) characterize the class of 1-perfectly orientable graphs within the classes
of cographs and of cobipartite graphs. The class of 1-perfectly orientable
co-bipartite graphs coincides with the class of co-bipartite circular arc
graphs.
