Boolean sum of graphs and reconstruction up to complementation

International audienceLet VVV be a set of cardinality vvv (possibly infinite). Two graphs GGG and G′G′G' with vertex set VVV are {\it isomorphic up to complementation} if G′G′G' is isomorphic to GGG or to the complement G¯¯¯¯G¯\overline G of GGG. Let kkk be a non-negative integer, GGG and G′G′G' are {\it kkk-hypomorphic up to complementation} if for every kkk-element subset KKK of VVV, the induced subgraphs G↾KG↾KG_{\restriction K} and G′↾KG↾K′G'_{\restriction K} are isomorphic up to complementation. A graph GGG is {\it kkk-reconstructible up to complementation} if every graph G′G′G' which is kkk-hypomorphic to GGG up to complementation is in fact isomorphic to GGG up to complementation. We prove that a graph GGG has this property provided that 4≤k≤v−34≤k≤v−34\leq k\leq v-3. Moreover, under these conditions, if $