Hypomorphy of graphs up to complementation

Let V be a set of cardinality v (possibly infinite). Two graphsG 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 subsetK 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 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. Ille [8]. MSC : 05C50; 05C60.