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 Gˉ 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_↾K and G′_↾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 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 S contains all pairs (n,k) such that 4≤k≤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. Ill