The Complement of a D-Tree is Pure Shellable

Let G be a simple undirected graph and let G be a simplicial complex whose faces correspond to the independent sets of G. A graph G is called shellable if G is a shellable simpli- cial complex. We prove that the complement of a d-tree is a pure shellable graph. This generalizes a recent result of Ferrarello who used a theorem due to R. Froberg to prove that the complement of a d-tree is a Cohen-Macaulay graph.