Monotone maximum partial-twuality widths of vf-safe delta-matroids

Abstract

For a delta-matroid, the maximum twist width theorem states that the maximum width over all twists can be reached along a non-decreasing sequence of intermediate twist widths. In this paper we study analogous monotone maximum width sequences for partial twualities generated by twist and loop complementation. We prove that, for each non-twist partial-twuality operation on a vf-safe delta-matroid, there exists a subset attaining the corresponding maximum partial-twuality width whose elements can be ordered so that the successive intermediate widths are non-decreasing. Together with the known twist case, this gives a monotone maximum width theorem for all five nontrivial partial-twuality operations on vf-safe delta-matroids. We also prove feasible-set attainment results for the operations $\ast\times$ and $\ast\times\ast$. Finally, we translate these results to ribbon graphs, obtaining monotone sequences for maximum partial-twuality Euler genera and spanning quasi-tree attainment results for the corresponding ribbon graph operations.