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The Morse complex of the wedge of two extended star graphs and a path

Shuma Komatsu

TL;DR

This work extends discrete Morse theory to compute the homotopy types of Morse complexes for graph constructions formed by extended star graphs and path wedges. It develops a framework based on strong collapses, Hasse diagrams, star clusters, and the Cluster Lemma to obtain explicit homotopy types: $\mathcal{M}(S_{1,n}) \simeq S^n$ and a case analysis for $\mathcal{M}(P_t \vee S_{0,n} \vee S_{0,l})$ yielding wedge sums of spheres with dimensions depending on $t=3u$, $3u+1$, or $3u+2$. The paper also provides results for $\mathcal{M}(P_t \vee S_{1,n} \vee S_{1,l})$ and illustrates with the example $\mathcal{M}(P_6 \vee S_{0,2} \vee S_{0,2}) \simeq S^8 \vee S^8 \vee S^8$. These findings generalize prior work on extended star graphs and offer a computational approach for Morse complexes of graph joins and wedges.

Abstract

In the work of C. Donovan and N. A. Scoville, the homotopy type of the Morse complex of the extended star graph which is obtained as the one-point union of n paths of length 2 was determined by using star clusters and Cluster Lemma. In this paper, we determine the homotopy type of the Morse complex of extended star graph consisting of a path of length 1 and n paths of length 2 by using strong collapses and Hasse diagram. Furthermore, we compute the homotopy type of the graph obtained by attaching the center vertices of two extended star graphs to different endpoints of a path by using star clusters and the Cluster Lemma.

The Morse complex of the wedge of two extended star graphs and a path

TL;DR

This work extends discrete Morse theory to compute the homotopy types of Morse complexes for graph constructions formed by extended star graphs and path wedges. It develops a framework based on strong collapses, Hasse diagrams, star clusters, and the Cluster Lemma to obtain explicit homotopy types: and a case analysis for yielding wedge sums of spheres with dimensions depending on , , or . The paper also provides results for and illustrates with the example . These findings generalize prior work on extended star graphs and offer a computational approach for Morse complexes of graph joins and wedges.

Abstract

In the work of C. Donovan and N. A. Scoville, the homotopy type of the Morse complex of the extended star graph which is obtained as the one-point union of n paths of length 2 was determined by using star clusters and Cluster Lemma. In this paper, we determine the homotopy type of the Morse complex of extended star graph consisting of a path of length 1 and n paths of length 2 by using strong collapses and Hasse diagram. Furthermore, we compute the homotopy type of the graph obtained by attaching the center vertices of two extended star graphs to different endpoints of a path by using star clusters and the Cluster Lemma.

Paper Structure

This paper contains 4 sections, 14 theorems, 16 equations, 20 figures.

Table of Contents

  1. introduction
  2. Preliminaries
  3. extended star graphs
  4. main theorem

Key Result

Proposition 2.11

Let $K, L$ be connected simplicial complexes each with at least one edge. Then, where $\mathcal{M}(K)*\mathcal{M}(L)$ denotes the join of $\mathcal{M}(K)$ and $\mathcal{M}(L)$.

Figures (20)

  • Figure 1: A simplicial complex $K$ and a vertex $v\in K$ at the left and the star $\mathrm{st}\,_K(v)$ of $v$ at the right.
  • Figure 2: A 2-dimension flag complex at the left and the star cluster of a simplex $\sigma$ at the right.
  • Figure 3: $\Delta_0$ is the union of the star of each arrow.
  • Figure 4: Match $\bigcup_{i=1}^n\{(a_i)b_i\}$ with $\bigcup_{i=1}^n\{(a_i)b_i\}\cup\{(c)a_1\}$.
  • Figure 5: $S_{1, 3}$. By Theorem \ref{['thm:3b']}, we see that $\mathcal{M}(S_{1, 3})\simeq S^3$.
  • ...and 15 more figures

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: cf. DLS22DS23
  • Definition 2.6: cf. DLS22DS23
  • Definition 2.7: cf. DLS22DS23
  • Remark 2.8
  • Definition 2.9: cf. DLS22DS23
  • Definition 2.10: cf. DLS22DS23
  • ...and 25 more