  The purpose of this paper is to characterize graphs that do not have a large
$K_{2,n}$-minor. As corollaries, it is proved that, for any given positive
integer $n$, every sufficiently large 3-connected graph with minimum degree at
least six, every 4-connected graph with a vertex of sufficiently high degree,
and every sufficiently large 5-connected graph must have a $K_{2,n}$-minor.
