For a graph G, the total k-cut complex \(\Delta ^t_k(G)\) , introduced by Bayer et al. (Disc Math 346(7), 2023) is the simplicial complex whose facets are the complements of independent vertex sets of size k in G. It is known that \(\Delta ^t_k(G)\) is vertex decomposable for all k if and only if G is chordal. In this paper, we study the Alexander dual of \(\Delta ^t_k(G)\) for some specific complexes. We prove that the Alexander dual of the total cut complexes of powers of a path graph is pure vertex decomposable. To establish this result, we characterize the minimal hitting sets of the family of independent vertices of size k in G. As a consequence, combined with a result of Bayer et al., we confirm two conjectures of Fröberg which state that the Stanley–Reisner rings of the total cut complexes of paths and squared paths are Cohen–Macaulay and have linear resolutions. Furthermore, for any cycle \(C_n\) of length n, we show that, unlike \(\Delta ^t_k(C_n)\) itself, its Alexander dual \(\Delta ^t_k(C_n)^\vee \) is pure vertex decomposable, implying that the Stanley–Reisner ring of \(\Delta ^t_k(C_n)\) has a linear resolution. We investigate vertex decomposability for the total cut complexes of certain relative augmented graphs obtained from a vertex decomposable graph. We determine the homotopy type of \(\Delta ^t_k(C_n)^\vee \) , as well as that of the Alexander dual of the total cut complexes of powers of a path. Finally, we prove pure vertex decomposability of the Alexander dual of the total cut complexes of certain complete multipartite graphs and then characterize higher powers of cycles for which the Alexander dual of their total cut complexes are pure vertex decomposable.
