Neighborhood Complexes of induced $k$-independent graphs

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.