  The well known Wagner's theorem states that a graph is a planar graph if and
only if it is $K_5$-minor-free and $K_{3,3}$-minor-free. Denote by $AT(G)$ the
Alon-Tarsi number of a graph $G$. We show that for any $K_{3,3}$-minor-free
graph $G$, $AT(G)\le 5$, there exists a matching $M$ and a forest $F$ such that
$AT(G-M)\le 4$ and $AT(G-E(F))\le 3$, extending the result on the Alon-Tarsi
number of $K_5$-minor-free graphs due to Abe, Kim and Ozeki.
