Abstract : The paper presents designs that allow detection of mixed effects when performing preliminary screening of the inputs of a scalar function of $d$ input factors, in the spirit of Morris' Elementary Effects approach. We introduce the class of $(d,c)$-cycle equitable designs as those that enable computation of exactly $c$ second order effects on all possible pairs of input factors. Using these designs, we propose a fast Mixed Effects screening method, that enables efficient identification of the interaction graph of the input variables. Design definition is formally supported on the establishment of an isometry between sub-graphs of the unit cube $Q_d$ equipped of the Manhattan metric, and a set of polynomials in $(X_1,\ldots, X_d)$ on which a convenient inner product is defined. In the paper we present systems of equations that recursively define these $(d,c)$-cycle equitable designs for generic values of $c\geq 1$, from which direct algorithmic implementations are derived. Application cases are presented, illustrating the application of the proposed designs to the estimation of the interaction graph of specific functions.