The Forest Filtration of a Graph

TL;DR

The paper defines and analyzes a filtration of graph-associated simplicial complexes $\mathcal{F}_d(G)$ whose $d=0$ case is the independence complex and whose $d=\infty$ case records forests. It develops a toolkit based on homotopy colimits, nerve theorems, and Alexander duality to compute the homotopy types of these complexes for key graph families (paths, cycles, double stars, cactus graphs) and under standard graph operations (joins and products). It then derives bounds on graph parameters such as the decycling number $\nabla_d(G)$ and the vertex cover number via rational cohomology, and provides Fibonacci-type bounds for ternary graphs using these topological invariants. The results yield explicit decompositions into wedges of spheres in many cases, clarify the topological structure under joins and products, and connect the graph-theoretic parameters to global topological features, offering both exact homotopy types and practical bounds for combinatorial optimization problems.

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.

Figures (2)

• Figure 1: $C_n+e$
• Figure 2:

Theorems & Definitions (84)

• Theorem 1: Nerve Theorem, see bjornertopmeth
• Theorem 2: Whitehead's theorem, see hatcher
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• Lemma 7
• proof
• Theorem 8
• proof