  We investigate families of two-dimensional simplicial complexes defined in
terms of vertex decompositions. They include nonevasive complexes, strongly
collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary
and Palmer. We investigate the complexity of recognition problems for those
families and some of their combinatorial properties. Certain results follow
from analogous decomposition techniques for graphs. For example, we prove that
it is NP-complete to decide if a graph can be reduced to a discrete graph by a
sequence of removals of vertices of degree 3.
