August 17, 2025

Vertex decompositions of two-dimensional complexes and graphs

Key Points

Certain families of two-dimensional simplicial complexes are defined by specific vertex decompositions, and their combinatorial properties are explored.
NP-completeness is established for determining if a graph can be reduced to a discrete graph by removing vertices of degree 3.
The exploration involves nonevasive and strongly collapsible complexes, highlighting their complex recognition problems.
Findings demonstrate connections between complex structures and analogous decomposition techniques in graph theory.

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.

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Cite This Study

Michał Adamaszek (Mon,) studied this question.

synapsesocial.com/papers/68a370e80a429f79733333fc https://doi.org/https://doi.org/10.55016/ojs/cdm.v7i2.61908

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