Claw-freeness, 3-homogeneous subsets of a graph and a reconstruction problem

We describe $\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{b}\{K_{1,3},\overline{K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $\bar{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths, 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 $G\dot{+}{G}^{\mathrm{\prime }}$ of two graphs $G$ and $G^{\mathrm{\prime }}$ giving the same hypergraph. We indicate the role of this latter description in a reconstruction problem of graphs up to complementation.