On The Complement Of An Ambisidigraph

More than forty years ago, F. Harary raised the problem of defining the complement Sc of a given signed graph S while discussing “structural duality” in general, arguing the concept to be a ‘philosophical necessity’ towards developing eventually a theoretical understanding of the structural dynamics of a social system. As we expect the concept to be of fundamental importance in probing dynamic states of structural stability of a social system arising in epochs of its evolution, here we continue the study of our recently introduced notion of the unary operation of obtaining the complement of a given sociogram, viz., a selfloop-free signed multidigraph in which every pair of distinct vertices is joined by at most one arc of each sign in each direction or what we have preferred to call here an ambisidigraph. Particularly, we discuss the problem of determining self-complementary sociograms as also unavoidability, in a well-defined sense, of balanced cycles in certain special classes of sociograms. In particular, we find a very interesting connection between Heiderʹs theory of social balance and the notion of complementation in social systems by showing that for any ambisidigraph S and for any positive integer k, either S and Sc have no (2k+ 1)-cycle or at least one of them has a balanced (2k + 1)-cycle.