The size of a graph is reconstructible from any n-2 cards

Abstract Let G and H be graphs of order n . The number of common cards of G and H is the maximum number of disjoint pairs ( v , w ) , where v and w are vertices of G and H , respectively, such that G − v ≅ H − w . We prove that if the number of common cards of G and H is at least n − 2 then G and H must have the same number of edges when n ≥ 29 . This is the first improvement on the 25 -year-old result of Myrvold that if G and H have at least n − 1 common cards then they have the same number of edges. It also improves on the result of Woodall and others that the numbers of edges of G and H differ by at most 1 when they have n − 2 common cards.