More on Vertex-Switching Reconstruction

A graph is called s-vertex switching reconstructible (s-VSR) if it is uniquely defined, up to isomorphism by the multiset of unlabeled graphs obtained by switching of all its s-vertex subsets. Stanley proved that a graph with n vertices is s-VSR if the Krawtchouk polynomial Pns has no even roots. Solving balance equations, introduced in Krasnikov and Roditty (Arch. Math. (Basel) 48 (1987). 458-464) for the switching reconstruction problem, we show that a graph is s-VSR if the corresponding Krawtchouk polynomial has one or two even roots laying far from n/2. As a consequence we prove that graphs with sufficiently large number n of vertices are s-VSR for some values of s about n/2. In particular, all graphs are s-VSR for n − 2s = 0, 1, 3. and if n ≠ 0 (mod 4), for n − 2s = 2, 6.