Vertex Decomposability of 2-CM and Gorenstein Simplicial Complexes of Codimension 3

Let Delta be a simplicial complex on vertex set [n]. It is shown that if Delta is complete intersection, Cohen-Macaulay of codimension 2, Gorenstein of codimension 3, or 2-Cohen-Macaulay of codimension 3, then Delta is vertex decomposable. As a consequence, we show that if A is a simplicial complex such that I-Delta = I-t (C-n), where I-t(C-n) is the path ideal of length t of C-n, then Delta is vertex decomposable if and only if t = n, t = n 1, or n is odd and t = (n 1)/2.