  Let $V$ be a set of cardinality $v$ (possibly infinite). Two graphs $G$ and
$G'$ with vertex set $V$ are {\it isomorphic up to complementation} if $G'$ is
isomorphic to $G$ or to the complement $\bar G$ of $G$. Let $k$ be a
non-negative integer, $G$ and $G'$ are {\it $k$-hypomorphic up to
complementation} if for every $k$-element subset $K$ of $V$, the induced
subgraphs $G\_{\restriction K}$ and $G'\_{\restriction K}$ are isomorphic up to
complementation. A graph $G$ is {\it $k$-reconstructible up to complementation}
if every graph $G'$ which is $k$-hypomorphic to $G$ up to complementation is in
fact isomorphic to $G$ up to complementation. We give a partial
characterisation of the set $\mathcal S$ of pairs $(n,k)$ such that two graphs
$G$ and $G'$ on the same set of $n$ vertices are equal up to complementation
whenever they are $k$-hypomorphic up to complementation. We prove in particular
that $\mathcal S$ contains all pairs $(n,k)$ such that $4\leq k\leq n-4$. We
also prove that 4 is the least integer $k$ such that every graph $G$ having a
large number $n$ of vertices is $k$-reconstructible up to complementation; this
answers a question raised by P. Ille
