The Topology of $k$-Robust Clique Complexes in Grid-like Graphs

TL;DR

This paper introduces and analyzes $k$-robust clique complexes in grid-like graphs, revealing their homotopy types as wedges of spheres and connecting these findings to total-$k$-cut complexes.

Contribution

It generalizes clique complexes to $k$-robust variants, characterizes their homotopy types in grid graphs, and links these to total-$k$-cut complexes using algebraic topology tools.

Findings

Homotopy type for $k=2$ and $k=3$ is a wedge of spheres.

Extended homotopy type results to arbitrary $k$ under structural constraints.

Connected homotopy types of total-$k$-cut complexes to $k$-robust clique complexes.

Abstract

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 investigate these complexes for square sequence graphs, a class of bipartite graphs introduced here that are constructed by iteratively attaching $C_4$ cycles. This class includes rectangular grid graphs $G_{m,n}$. We show that for $k=2$ and $k=3$, the homotopy type is a wedge sum of $(2k-3)$-dimensional spheres, a result we extend to arbitrary $k$ under specific structural constraints on the attachment sequence. Our approach utilizes K\"{o}nig's theorem to decompose the complex into manageable components, whose homotopy types are easy to understand. This then enables an inductive proof based on the decomposition and standard tools of algebraic topology. Finally, we utilize Alexander duality to connect our results to the study of total-$k$-cut complexes, generalizing recent results concerning the homotopy types of total-$k$-cut complexes for grid graphs.