3-diagonal equation and planarity of graphs

In his PhD thesis, S. Krstic used (multi)graphs to solve generalized quadratic quasigroup functional equations. In particular, he showed the fundamental role of Kuratowski Theorem on planarity of graphs in determining properties of general solutions of such equations. As a first step towards generalization of his results to functional equations on ternary quasigroups, we consider generalized 3-diagonal equation A(B(x, y, z),C(y, u, v),D(z, v,w)) = E(x, u,w). This is one of equations with complete graph K5 as a corresponding graph. General solution of this equation is given, confirming the important role of Kuratowski Theorem in this case as well.