Hypomorphy of graphs up to complementation

Let V be a set of cardinality v (possibly infinite). Two graphs G and G ′ with vertex set V are isomorphic up to complementation if G ′ is isomorphic to G or to the complement G ¯ of G. Let k be a non-negative integer, G and G ′ are 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 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 ordered 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 ordered 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. Ille [P. Ille, Personal communication, September 2000].