  Gross, Mansour and Tucker introduced the partial-twuality polynomial of a
ribbon graph. Chumutov and Vignes-Tourneret posed a problem: it would be
interesting to know whether the partial duality polynomial and the related
conjectures would make sense for general delta-matroids. In this paper we
consider analogues of partial-twuality polynomials for delta-matroids. Various
possible properties of partial-twuality polynomials of set systems are studied.
We discuss the numerical implications of partial-twualities on a single element
and prove that the intersection graphs can determine the partial-twuality
polynomials of bouquets and normal binary delta-matroids, respectively.
Finally, we give a characterization of vf-safe delta-matroids whose
partial-twuality polynomials have only one term.
