The Structure Of Graphs With No K3,3 Immersion

The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either K3,3 or K5 as a minor. Using this, Wagner obtained a precise description of all graphs with no K3,3-minor and all graphs with no K5-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 K3,3 as an immersion. This strengthens an earlier theorem of Giannopoulou, Kaminski, and Thilikos that gives a rough description of the class of graphs with no K3,3 or K5 immersion.