A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs

TL;DR

The paper investigates the complex $\mathcal{N}_n$ of disconnected graphs on $n$ labeled vertices, whose known homotopy type is a wedge of $$(n-1)!$$ spheres of dimension $$(n-3)$$. It applies Forman's discrete Morse theory to construct a discrete gradient vector field $\mathscr{V}$ on $\mathcal{N}_n$ and derives the homotopy type under the gradient assumption. The authors identify a pivotal claim used to justify acyclicity as invalid, offering counterexamples and then providing an alternative general argument proving $\mathscr{V}$ is indeed a gradient vector field. The work also presents a robust, broadly applicable technique for verifying acyclicity in discrete vector-field constructions, highlighting the practical impact of combinatorial Morse theory in determining topological structure.

Abstract

Robin Forman's highly influential 2002 paper A User's Guide to Discrete Morse Theory presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract simplicial complex of disconnected graphs of order $n$ (which was previously done by Victor Vassiliev using classical topological methods) using discrete Morse theoretic techniques, which are purely combinatorial in nature. The techniques involve the construction (and verification) of a discrete gradient vector field on the complex. However, the verification part relies on a claim that doesn't seem to hold. In this note, we provide a couple of counterexamples against this specific claim. We also provide an alternative proof of the bigger claim that the constructed discrete vector field is indeed a gradient vector field. Our proof technique relies on a key observation which is not specific to the problem at hand, and thus is applicable while verifying a constructed discrete vector field is a gradient one in general.

Figures (3)

• Figure 1: The $2$-simplex $\alpha_0$ is paired off with the $3$-simplex $\beta_0$ in $\mathscr{V}_3\setminus \mathscr{V}_{12}$, and the $2$-simplex $\alpha_1$ is paired off with the $3$-simplex $\beta_1$ in $\mathscr{V}_4\setminus \mathscr{V}_3$.
• Figure 2: The $1$-simplex $\alpha_0$ is paired off with the $2$-simplex $\beta_0$ in $\mathscr{V}_3\setminus \mathscr{V}_{12}$. Both choices of $\alpha_1$ are unpaired in $\mathscr{V}$.
• Figure 3: The graph $\alpha_0$ ($\in \mathcal{N}_{11}$) first gets paired off with $\alpha_0+(4,7)$ in $\mathscr{V}_7$, and thus, $k=7$, $j=4$. Here, $F$ is the subgraph induced by $\{1,\ldots,6\}$ (bounded by dashed (blue) lines), and $H_0$ is the subgraph induced by $\{4,7,8,10\}$ (bounded by solid (red) lines).

Theorems & Definitions (6)

• Theorem 1.1
• Definition 2.1: Discrete vector field
• Definition 2.2: Gradient vector field
• Definition 2.3: Critical simplex
• Theorem 2.4
• Claim 4.1: forman02