Switching reconstruction and diophantine equations (Q914705)

Based on a result of \textit{R. P. Stanley} [``Reconstruction from vertex switching'', J. Comb. Theory, Ser. B 38, 132-138 (1985; Zbl 0572.05046)], we show that for each \(s\geq 4\) there exists an integer \(N_ s\) such that any graph with \(n>N_ s\) vertices is reconstructible from the multiset of graphs obtained by switching of vertex subsets with s vertices, provided \(n\neq O(mod 4)\) if s is odd. We also establish an analog of \textit{P. J. Kelly}'s lemma [``A congruence theorem for trees'', Pac. J. Math. 7, 961-968 (1957; Zbl 0078.371)] for the above s-switching reconstruction problem.