Partial-twuality polynomial interpolation for binary delta-matroids

Abstract

Gross, Mansour and Tucker introduced the partial-twuality polynomials for ribbon graphs and investigated the interpolation property of these polynomials. The ribbon group generated by $δ$ and $τ$ acts on set systems as twist $\ast$ and loop complementation $\times$, yielding five nontrivial twuality operators: $ \{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast \}.$ Yan and Jin extended partial-twuality polynomials to set systems, yielding partial-$\bullet$ polynomials with $\bullet\in\{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast\}.$ For partial-$\ast$ polynomials, Zhao and Yan proved that this polynomial is either even, odd, or both even-interpolating and odd-interpolating for every binary delta-matroid. In this paper, we extend this interpolation property to all the remaining nontrivial partial-twualities of binary delta-matroids. Consequently, for every binary delta-matroid and every $\bullet\in\{\ast,\times,\ast \times ,\times\ast ,\ast \times \ast \}$, the partial-$\bullet$ polynomial is either even, odd, or both even-interpolating and odd-interpolating. We also provide examples to show that the binary assumption is essential.