The graphs with only self-dual signings

Abstract

Given a graph G, it is possible to attach positive and negative signs to its lines only, to its points only, or to both. The resulting structures are called respectively signed graphs, marked graphs and nets. The dual of each such structure is obtained by changing every sign in it. We determine all graphs G for which every suitable marked graph on G is self-dual (the M-dual graphs), and also the corresponding graphs G for signed graphs (S-dual) and for nets (N-dual.A graph G is M-dual if and only if G or G is one of the graphs K2m, 2Km, mK2, Km + K2 or 2C4. The S-dual graphs are C6, 2C3, 2C4, 2K1n, 2nK2, K1,2n, nK1,2, K2n, Kn and all graphs obtained from these by the addition of isolated points. Finally, the only N-dual graph other than -K2n is 2K2.