Switching reconstruction of signed graphs: net-degree and balance

The switching reconstruction problem for $2$-edge-colored graphs, introduced by Bondy and Mercier, asks whether such a graph is determined up to isomorphism by the multiset of the graphs obtained by exchanging the colors of the edges at each vertex; for complete underlying graphs it is the vertex-switching reconstruction problem of Stanley. We study this problem in the language of signed graphs, where the net-degree of a vertex and the notion of balance come naturally into play. We prove that a signed graph on n≠4 vertices is reconstructible when its net-degrees are all positive, all negative or all zero, which generalizes the reconstructibility of regular graphs in Stanley's problem. For connected balanced signed graphs, whose switching reconstruction problem we show to be equivalent to that of connected graphs with a distinguished bipartition of the vertex set, we prove reconstructibility when the two parts of the bipartition differ in size by more than two, and when all net-degrees are non-negative.