The [formula omitted]-hypomorphy of digraphs up to complementation

Abstract

Two digraphs G=(V,E) and G′=(V,E′) are isomorphic up to complementation if G′ is isomorphic to G or to the complement G¯≔(V,{(x,y)∈V2:x≠y,(x,y)∉E}) of G. The Boolean sum G+̇G′ is the symmetric digraph U=(V,E(U)) defined by {x,y}∈E(U) if and only if (x,y)∈E and (x,y)∉E′, or (x,y)∉E and (x,y)∈E′. Let k be a nonnegative integer. The digraphs G and G′ are (≤k)-hypomorphic up to complementation if for every t-element subset X of V, with t≤k, the induced subdigraphs G↾X and G↾X′ are isomorphic up to complementation. The digraphs G and G′ are hereditarily isomorphic (resp. hereditarily isomorphic up to complementation) if for each subset X of V, the induced subdigraphs G↾X and G↾X′ are isomorphic (resp. isomorphic up to complementation). Here, we give the form of the pair {G,G′} whenever G and G′ are two digraphs, (≤5)-hypomorphic up to complementation and such that the Boolean sum U≔G+̇G′ and the complement U¯ are both connected, and thus we deduce that G and G′ are hereditarily isomorphic up to complementation; we prove also that the value 5 is optimal. The case U or U¯ is not connected is studied in a forthcoming paper.