Incremental construction properties in dimension two-- shellability, extendable shellability and vertex decomposability

We give new examples of shellable but not extendably shellable two dimensional simplicial complexes. They include minimal examples, which are smaller than those previously known. We also give examples of shellable but not vertex decomposable two dimensional simplicial complexes. Among them are extendably shellable ones. This shows that neither extendable shellability nor vertex decomposability implies the other. We found these examples by enumerating shellable two dimensional simplicial complexes which are not pseudomanifolds. A rather e cient algorithm for this enumeration is also given.