A Characterization of Graphs with No Cube Minor

In this paper it is shown that any 4-connected graph that does not contain a minor isomorphic to the cube is a minor of the line graph of Vn for some n⩾6 or a minor of one of five graphs. Moreover, there exists a unique 5-connected graph on at least 8 vertices with no cube minor and a unique 4-connected graph with a vertex of degree at least 8 with no cube minor. Further, it is shown that any graph with no cube minor is obtained from 4-connected such graphs by 0-, 1-, and 2-summing, and 3-summing over a specified triangles.