A note on α-drawable k-trees

We study the problem of realizing a given graph as an α-complex of a set of points in the plane. The graphs we consider are trees and 2-trees. In the case of 2-trees, we confine our attention to the realizability of graphs as the α-complex minus faces of dimension two; in other words, realizability of the graph in terms of the 1-skeleton of the α-complex of the point set. We obtain both positive (realizability) and negative (non-realizability) results.