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

Marek Filakovský

TL;DR

The paper studies the topology of $k$-robust clique complexes $\mathsf{Cliq}_{k}(G)$ on grid-like graphs, introducing a square sequence framework and proving an inductive decomposition based on König's theorem. For $k=2,3$, it shows the homotopy type is a wedge of $(2k-3)$-spheres with multiplicities given by $\binom{(m-1)(n-1)}{k-1}$, and extends the analysis to general $k$ under structural constraints. Through Alexander duality, these results yield corresponding topological descriptions for total-$k$-cut complexes $\Delta_k^t(G)$, linking robust clique topology to cut-complex topology. The approach provides an explicit, inductive, and tractable description of the homotopy types on grid graphs, highlighting a dual relationship between robust clique complexes and total cut 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 König'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.

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

TL;DR

The paper studies the topology of -robust clique complexes on grid-like graphs, introducing a square sequence framework and proving an inductive decomposition based on König's theorem. For , it shows the homotopy type is a wedge of -spheres with multiplicities given by , and extends the analysis to general under structural constraints. Through Alexander duality, these results yield corresponding topological descriptions for total--cut complexes , linking robust clique topology to cut-complex topology. The approach provides an explicit, inductive, and tractable description of the homotopy types on grid graphs, highlighting a dual relationship between robust clique complexes and total cut complexes.

Abstract

We introduce -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 . We investigate these complexes for square sequence graphs, a class of bipartite graphs introduced here that are constructed by iteratively attaching cycles. This class includes rectangular grid graphs . We show that for and , the homotopy type is a wedge sum of -dimensional spheres, a result we extend to arbitrary under specific structural constraints on the attachment sequence. Our approach utilizes König'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--cut complexes, generalizing recent results concerning the homotopy types of total--cut complexes for grid graphs.
Paper Structure (3 sections, 15 theorems, 45 equations, 2 figures)

This paper contains 3 sections, 15 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of main results

Key Result

Theorem 1.1

Let $G_{m,n}$ be an $m \times n$ grid graph with $m, n \geq 2$. For $k\in \{2,3\}$,

Figures (2)

  • Figure 1: Edge and corner gluing
  • Figure 2: The robust $3$-clique complex of the graph on the left consists of three $2$-simplices.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Definition 2.1
  • Definition 2.2: Grid graph
  • Definition 2.3
  • Definition 2.4
  • Example
  • ...and 24 more