A note on highly connected $K_{2,\mathrm{\ell }}$-minor free graphs

We show that every $3$-connected $K_{2,\mathrm{\ell }}$-minor free graph with minimum degree at least $4$ has maximum degree at most $7\mathrm{\ell }$. As a consequence, we show that every 3-connected $K_{2,\mathrm{\ell }}$-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 $K_{2,\mathrm{\ell }}$-minor free graphs.