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

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.