Vertex decompositions of two-dimensional complexes and graphs

TL;DR

This paper explores vertex decomposition methods for two-dimensional complexes and graphs, analyzing their properties, recognition complexity, and relationships to known classes like nonevasive and collapsible complexes.

Contribution

It introduces new families of 2D complexes based on vertex decompositions and studies their recognition complexity and combinatorial properties.

Findings

Recognition of certain complexes is NP-complete.

Vertex decompositions relate complexes to graph reduction problems.

Analogies between complexes and graphs are established.

Abstract

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.