Spectral Algorithms and Representations April 2005 The Colin de Verdière Number and Nullspace Embeddings

Loosely speaking, an embedding of a graph G in Rr consists of an injective mapping ψ : V → Rr, and a correspondence from each edge ij ∈ E to a simple curve in Rr with endpoints ψ(i) and ψ(j). Here a curve can be taken to be the image of a continuous injective function φ : [0, 1] → Rr. An embedding in S2 is defined similarly. In a planar embedding images of edges cross only at images of vertices shared by the edges. A graph is planar if it has a planar embedding, outerplanar if it can be planarly embedded so that all the vertices are on the outer face, and linklessly embeddable if it can be embedded in R3 so that any two disjoint circuits form unlinked curves in R3 (see Figure 1 for an illustration). These notes assume familiarity with embeddings and planarity. More details on planar graphs can be found in standard combinatorial optimization and graph theory textbooks, e.g. [KV02, Die00, Bol98].