Applications of balance equations to vertex switching reconstruction

Abstract

AbstractA 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. We show that a graph with n vertices is n/2‐VSR if n = 0(mod 4), (n − 2)/2‐VSR and n/2‐VSR if n = 2(mod 4), (n − 1)/2‐VSR if n = 1 (mod 2). For hypothetical nonreconstructible graphs, we give bounds on the number of edges (for any s) and on the maximum and minimum degree (for s = 2). We also show that for n > 9 the degree sequence is 2‐VSR.