  The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if
it does not have either $K_{3,3}$ or $K_5$ as a minor. Using this Wagner
obtained a precise description of all graphs with no $K_{3,3}$ minor and all
graphs with no $K_5$ minor. Similar results have been achieved for the class of
graphs with no $H$-minor for a number of small graphs $H$.
  In this paper we give a precise structure theorem for graphs which do not
contain $K_{3,3}$ as an immersion. This strengthens an earlier theorem of
Giannopoulou, Kami\'{n}ski, and Thilikos that gives a rough description of the
class of graphs with no $K_{3,3}$ or $K_5$ immersion.
