The excluded minors for $k$-polymatroids with binary $k$-natural matroids

Abstract

If $\mathcal{C}$ is a minor-closed class of matroids, then the class $\widetilde{\mathcal{C}}'_k$ of $k$-polymatroids whose $k$-natural matroids are in $\mathcal{C}$ is also minor-closed. We investigate the following question: When $\mathcal{C}$ is the class of binary matroids, what are the excluded minors for $\widetilde{\mathcal{C}}'_k$? When $k = 1$, $\widetilde{\mathcal{C}}'_1$ is simply the class of binary matroids, which has $U_{2,4}$ as its only excluded minor. Joseph E. Bonin and Kevin Long answered the question for $k = 2$ and found that the set of excluded minors for $\widetilde{\mathcal{C}}'_2$ is infinite. We determine the sets of excluded minors for $\widetilde{\mathcal{C}}'_k$ when $k \geq 3$ and find that they are finite. There are $12$ excluded minors for $\widetilde{\mathcal{C}}'_3$ and when $k > 3$, there are $k+7$ excluded minors for $\widetilde{\mathcal{C}}'_k$.