  We describe $Forb\{K_{1,3}, \overline {K_{1,3}}\}$, the class of graphs $G$
such that
  $G$ and its complement $ \overline{G}$ are claw-free. With few exceptions, it
is made of graphs whose connected components consist of cycles of length at
least 4, paths or isolated vertices, and of the complements of these graphs.
Considering the hypergraph ${\mathcal H} ^{(3)}(G)$ made of the 3-element
subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an
independent subset, we deduce from above a description of the Boolean sum
$G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate
the role of this latter description in a reconstruction problem of graphs up to
complementation.
