Related papers: Independence Complexes of Hexagonal Grid Graphs
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…
For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine 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.
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 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…
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.
It was conjectured by Goyal, Shukla and Singh that the independence complex of the categorical product $K_2\times K_3\times K_n$ has the homotopy type of a wedge of $(n-1)(3n-2)$ spheres of dimension $3$. Here we prove this conjecture by…
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 $P_n\times P_m$,…
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…
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…
The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$. Braun and Hough introduced a…
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…
The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun…
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…
We study the independence complex of the lexicographic product $G[H]$ of a forest $G$ and a graph $H$. We prove that for a forest $G$ which is not dominated by a single vertex, if the independence complex of $H$ is homotopy equivalent to a…
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…
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…
We provide a recursive construction of an acyclic matching (also known as a gradient vector field, an equivalent notion to a discrete Morse function) on the independence complex of a graph with a simplicial vertex using given acyclic…
We introduce the notion of \textit{star cluster} of a simplex in a simplicial complex. This concept provides a general tool to study the topology of independence complexes of graphs. We use star clusters to answer a question arisen from…
We introduce $k$-robust clique complexes, a family of simplicial complexes that generalizes the traditional clique complex. Here, a subset of vertices forms a simplex provided it does not contain an independent set of size $k$. We…