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 $K_{2,\ell}$-无 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 $5$ 且无 $5$ 度孪生点的 $3$-连通 $K_{2,\ell}$-无 minors 图具有有界大小。我们的证明使用了 Steiner 树和嵌套割集；特别地，它们不依赖于 Ding 对 $K_{2,\ell}$-无 minors 图的刻画。