Topology of cut complexes of graphs (Q6552470)

This pretty extensive article, which comprises 46 pages, studies an analogue of the Alexander dual of a simplicial complex. Given a positive integer \(k\), the \(k\)-cut complex \(\Delta_k (G)\) of a graph \(G\) is defined to be a simplicial complex with the vertex set \(V(G)\) and facets defined by separating sets of \(G\) with size \(|V(G)| - k\). For example, if \(G = C_5\), a cycle of length \(5\), then \(\Delta_2 (G)\) is the unique triangulation of the Möbius Strip with five vertices.\N\NThe main results of this article are related to the following eight specific families of graphs, namely, (1) complete bipartite graphs, (2) complete multipartite graphs, (3) cycles, (4) squared cycles, (5) \(K_n \times K_2\), (6) trees, (7) threshold graphs, and (8) connected triangle-free graphs. For the corresponding \(k\)-cut complexes, the following three questions are addressed: (A) Are they shellable? (B) What are their homotopy types and Betti numbers? (C) What is their equivariant homology? Therefore, totally \(8 \times 3 = 24\) questions have been raised. Many of the questions, but not all, are answered in the article. For the unanswered questions, several conjectures have been put forward. The authors use methods of algebraic topology, discrete Morse theory, and equivariant poset topology. \N\NThe following is an example of the results of the article for the squared cycle \(W_n\) with \(n\) vertices: The \(3\)-cut complex \(\Delta_k (W_{k+4})\) is homotopy equivalent to the circle \(\mathbb{S}^1\) and thus is not shellable. Two related conjectures are posed: (I) For \(k \geqslant 3\), the cut complex \(\Delta_k (W_n)\) is shellable for \(n \geqslant k + 6\), and (II) for \(k = 3\) and \(n \geqslant 9\), the Betti numbers are \(\binom{n-4}{2} - 9 = \{ 1, 6, 12, 19, 27, \ldots \}\) (this sequence is known as OEIS A051936). By now, both conjectures have been proved in case \(k=3\) by \textit{P. Chauhan} et al. [``$3$-Cut Complexes of Squared Cycle Graphs'', Preprint, \url{arXiv:2406.01979}].