Our aim in this paper is to continue our investigation of the homology of deleted products of finite, contractible, 2-dimensional polyhedra. In [1], we observed that if X is such a polyhedron, then a homeomorph of X can be constructed by starting with a 2-simplex and appending n-simplexes (n = 1, 2). In this paper, we are concerned with those polyhedra which have the property that if they are constructed as above, then at some stage we are forced to add to X i–1 a 2-simplex τ at two of its 1-faces, 〈u 3, u 1〉 and 〈u 3, u 2〉, where there is a simple closed curve S in ∂(St(u 3, X i–1)) such that u 1 and u 2 are not in S but every sequence of 1-simplexes in ∂(St(u 3, X i–1)) from u 1 to u 2 intersects S. The cone over the complete graph on five vertices and the cone over the houses-and-wells figure are examples of such spaces.