We show that every $3$-connected $K_{2,\ell}$-minor free graph with minimum degree at least $4$ has maximum degree at most $7\ell$. As a consequence, we show that every 3-connected $K_{2,\ell}$-minor free graph with minimum degree at least $5$ and no twins of degree $5$ has bounded size. Our proofs use Steiner trees and nested cuts; in particular, they do not rely on Ding's characterization of $K_{2,\ell}$-minor free graphs.

翻译：我们证明了每个最小度至少为4的3-连通无$K_{2,\ell}$- minors图的最大度至多为$7\ell$。作为推论，我们证明了每个最小度至少为5且不含5度孪生点的3-连通无$K_{2,\ell}$- minors图具有有界规模。我们的证明使用了斯坦纳树和嵌套割集；特别地，该证明不依赖于Ding关于无$K_{2,\ell}$- minors图的刻画。