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

Author(s): Maurice Pouzet | Hamza Si Kaddour | Nicolas Trotignon

Journal: Contributions to Discrete Mathematics
ISSN 1715-0868

Volume: 6;
Issue: 1;
Date: 2011;
Original page

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 $Gdot{+}G'$ 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.

Tango Jona
Tangokurs Rapperswil-Jona

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