Neighborhood Complexes of induced $k$-independent graphs

TL;DR

This paper explores the topological properties of neighborhood complexes of induced $k$-independent graphs, revealing potential relationships with total $k$-cut complexes through algebraic topology and discrete Morse theory.

Contribution

It introduces new insights into the homotopy types of these complexes and their connections, expanding understanding of graph topological invariants.

Findings

Homotopy types of certain total cut complexes determined

Relationships between neighborhood complexes and total $k$-cut complexes suggested

Techniques from algebraic topology and Morse theory applied

Abstract

This paper is devoted to the neighborhood complexes of the induced $k$-independent graphs. Inspired by the surprising correspondence between total $k$-cut complex of $n$-cycle $C_n$ and neighborhood complex of stable Kneser graph $SG(n,k)$, we anticipate that the homotopy type of total cut complexes may have some relationships with the neighborhood complexes of induced $k$-independent graphs. We investigated the homotopy type of some total cut complexes and neighborhood complexes of some other graphs, using techniques from algebraic topology and discrete Morse theory.