Vertex decompositions of two-dimensional complexes and graphs

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.Comment: Improved presentation and fixed some bug

Clique complexes and graph powers

We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.Comment: V3: final versio

A new approach to Whitehead's asphericity question

We investigate Whitehead's asphericity question from a new perspective, using results and techniques of the homotopy theory of finite topological spaces. We also introduce a method of reduction to investigate asphericity based on the interaction between the combinatorics and the topology of finite spaces.Comment: 9 pages, 5 figure

Contractibility of the orbit space of the pp-subgroup complex via Brown-Forman discrete Morse theory

We give a simple proof that the orbit space of the $p$-subgroup complex of a finite group is contractible using Brown-Forman discrete Morse theory. This result was originally conjectured by Webb and proved by Symonds