Some connectivity properties for excluded minors of the graph invariant ν(G)

Let ν(G) be the graph invariant introduced by Colin de Verdière in J. Combin. Theory Ser. B. 74 (1998) 121. Let k≥0, and let H be an excluded minor of the class of graphs G with ν(G)≤k. We show that H has no vertex cuts of size at most two and that, if S is a vertex cut of size three of H, then G−S has two components, and S is the neighbourhood of a vertex v and the subgraph induced by S∪{v} is isomorphic to one of the graphs in a certain collection of six graphs.