Free products, principal sums, bipartition complete matroids and the linking operation $({\mathcal{M}}_{SP}\vee {\mathcal{M}}_{PQ})\times (S\uplus Q)$

Abstract

In this paper we examine the special class of bipartition complete matroids, which includes free products and principal sums of matroids (Crapo and Schmitt, 2005; Bonin and Kung, 2015). A matroid ${\mathcal{M}}_{SQ}$ on $S\uplus Q$ is said to be $\{S,Q\}$--complete iff whenever $b_S^1\uplus b_Q^1,b_S^1\uplus b_Q^2,$ and $b_S^2\uplus b_Q^1$ are bases of ${\mathcal{M}}_{SQ},$ we also have $b_S^2\uplus b_Q^2$ as a base of ${\mathcal{M}}_{SQ}.$ We use the linking operation ${\mathcal{M}}_{SP}{\leftrightarrow }_{\mathcal{M}}{\mathcal{M}}_{PQ}:\equiv ({\mathcal{M}}_{SP}\vee {\mathcal{M}}_{PQ})\times (S\uplus Q)$ (Theja and Narayanan, 2014) for this study. We construct the $\{S,Q\}$--augmentation of an arbitrary matroid and show that this augmentation is again a matroid, that the dual of the augmentation is the augmentation of the dual matroid and that the $\{S,Q\}$--augmentation of an $\{S,Q\}$--complete matroid leaves it unchanged. We also show that bipartition completeness is stable under minors in the appropriate bipartitioned sense. Finally, we show that free products and principal sums of matroids ${\mathcal{M}}_S,{\mathcal{M}}_Q$ are $\{S,Q\}$--complete and give explicit minimal decompositions for them.