Related papers: Higher Independence Complexes of graphs and their …

The independence complex of a graph is a simplicial complex whose faces correspond to the independent sets of $G$. While independence complexes have been studied extensively for many graph classes, including square grid graphs, relatively…

The independence complex of a graph G is a simplicial complex whose simplices are the independent sets in G. In the last couple of decades, the independence complexes of square grids (with various boundary conditions) have gained much…

In this article, we define a family of regular bipartite graphs and show that the homotopy type of the independence complexes of this family is the wedge sum of spheres of certain dimensions.

We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the…

For $r\geq 1$, the $r$-independence complex of a graph $G$, denoted Ind$_r(G)$, is a simplicial complex whose faces are subsets $A \subseteq V(G)$ such that each component of the induced subgraph $G[A]$ has at most $r$ vertices. In this…

We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the…

For $r\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is a simplicial complex whose faces are the subsets $H \subseteq E(G)$ of the edge set of $G$ such that the degree of any vertex in the induced subgraph $G[H]$ is at…

We show that if a graph $G$ involves a certain square grid graph as a full subgraph, then a certain operation on it yields a simplicial suspension of the independence complex of $G$. This generalizes a result of Csorba. As a corollary, we…

Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground…

This paper investigates the shellability of $r$-independence complexes $\mathcal{I}_r(G)$, a generalization of classical independence complexes introduced by Paolini and Salvetti. For a graph $G$, a subset $A \subseteq V(G)$ is…

The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we…

The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double…

Let $G$ be a graph and $r \ge 1$. A vertex subset is $r$-independent if every connected component of its induced subgraph has size at most $r$. The family of all such subsets forms a simplicial complex, the $r$-independence complex…

We study the independence complexes of graph products where at least one factor is a path. We also analyze the complexes of their induced subgraphs. We determine the homotopy type of the independence complex of the graphs $P_n\times P_m$,…

Independence complexes of circle graphs are purely combinatorial objects. However, when constructed from some diagram of a link $L$, they reveal topological properties of $L$, more specifically, of its Khovanov homology. We analyze the…

We determine the homotopy types of the independence complexes of $(n \times 4)$ and $(n \times 5)$-square grid graphs. In fact, they are homotopy equivalent to wedges of spheres.

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of…

We determine the homotopy types of the independence complexes of the $(n \times 6)$-square grid graphs. In fact, we show that these complexes are homotopy equivalent to wedges of spheres.

We provide lower bounds on the connectivity of the independence complexes of hypergraphs. Additionally, we compute the homotopy types of the independence complexes of $d$-uniform properly-connected triangulated hypergraphs.

In this article, we introduce the notion of a wedge of graphs and provide detailed computations for the independence complex of a wedge of path and cycle graphs. In particular, we show that these complexes are either contractible or wedges…