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

TL;DR

This paper determines the homotopy types of Morse complexes for specific extended star graphs and their combinations with paths, using advanced topological methods.

Contribution

It introduces new techniques to compute homotopy types of Morse complexes for extended star graphs and their unions with paths.

Findings

Homotopy type of Morse complex of a path and extended star graphs determined.

Homotopy type of combined graphs obtained by attaching star graphs to paths computed.

Application of strong collapses and Hasse diagrams in topological analysis.

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.