The Forest Filtration of a Graph

TL;DR

This paper introduces a graph filtration based on independence and acyclic sets, analyzes its properties, computes homotopy types for various graphs, and establishes bounds related to graph parameters and Fibonacci numbers.

Contribution

It defines a new graph filtration, studies its topological properties, and connects these to graph invariants like the decycling number and Fibonacci bounds.

Findings

Homotopy types computed for specific graph families

Upper bounds established for decycling number and Fibonacci numbers

Generalizations of graph parameters using cohomology dimensions

Abstract

Given a graph $G$, we define a filtration of simplicial complexes associated to $G$, $\mathcal{F}_0(G)\subseteq\mathcal{F}_1(G)\subseteq\cdots\subseteq\mathcal{F}_\infty(G)$ where the first complex is the independence complex and the last the complex is formed by the acyclic sets of vertices. We prove some properties of this filtration and we calculate the homotopy type for various families of graphs. We give an upper bound for the decycling number and generalizations of this parameter using the dimensions of the rational cohomology groups of these complexes. We also derive an upper bound for the Fibonacci numbers of ternary graphs.