On a Minor-Monotone Graph Invariant

For any undirected graph G = (V, E) let λ(G) be the largest d for which there exists a d-dimensional subspace X of RV with the property that for each nonzero x ∈ X, the positive support of x induces a nonempty connected subgraph of G. (Here the positive support of x is the set of vertices v with x(v) > 0.) We show that λ(G) is monotone under taking miners and clique sums. Moreover, we show that λ(G) ≤ 3 if and only if G has no K5- or V8-minor; that is, if and only if G arises from planar graphs by taking clique sums and subgraphs.