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

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 paper proves that for an m×n grid and k=2,3, the robust clique complex is homotopy equivalent to a wedge of \(\binom{(m-1)(n-1)}{k-1}\) spheres of dimension \(2k-3\).
• The paper uses König’s theorem, matching theory, embedded joins, and suspension arguments to analyze edge and corner square attachments through recursive decompositions.
• The results transfer via Alexander duality to total-k-cut complexes, while examples for larger k show that the binomial formula can fail near the graph’s independence number and motivate broader conjectures.

Overview

This paper introduces and studies $k$-robust clique complexes, a family of simplicial complexes attached to a graph $G$ in which a vertex set $W$ forms a simplex whenever the induced subgraph $G[W]$ contains no independent set of size $k$. The case $k=2$ recovers the ordinary clique complex; larger $k$ relaxes the clique condition by tolerating missing edges so long as they do not form a forbidden independent set. The author, Marek Filakovský, analyzes these complexes for square sequence graphs — bipartite graphs built by iteratively attaching 4-cycles via edge gluing or corner gluing — a class that contains all rectangular grid graphs $G_{m,n}$. The main results determine homotopy types for $k \in \{2,3\}$ as wedge sums of spheres of dimension $2k-3$, extend to arbitrary $G$0 under pure edge-gluing sequences, and transfer to total-$G$1-cut complexes via Alexander duality.

The central methodological contribution is a decomposition lemma derived from König's theorem (via Gallai's identity $G$2) and Berge's lemma on maximum matchings. This decomposes $G$3 for a square sequence graph into a union $G$4, where $G$5 and $G$6 is an embedded join with the newly attached square. The embedded join $G$7, borrowed from commutative algebra contexts [SimisU2001, ChoeJ2025], behaves like the standard join up to homotopy when the factors intersect in a full simplex; the author notes this may be its first appearance in the combinatorics of simplicial complexes.

Main results

The principal theorem states that for an $G$8 grid graph with $G$9 and $W$0,

$W$1

Thus the number of top-dimensional holes equals the number of $W$2-subsets of the squares of the grid, and the dimension grows linearly in $W$3. For $W$4 this follows from the elementary observation that the clique complex of a triangle-free square sequence graph is the graph itself, which deformation retracts onto a spanning tree, leaving one circle per attached square.

For $W$5, the result follows from a recursive characterization (Theorem 2): if $W$6 has square sequence $W$7, then $W$8, where $W$9 satisfies a recurrence determined by the attachment mode:

Attachment	Recurrence for $G[W]$0
Edge gluing	$G[W]$1
Corner gluing	$G[W]$2

Here $G[W]$3 is the set of common neighbors of the two attachment vertices $G[W]$4 in $G[W]$5. For grid graphs glued "row by row," corner attachments always have $G[W]$6, so both cases yield the same increment $G[W]$7 at step $G[W]$8, producing the binomial coefficient $G[W]$9.

A corollary extends the enumerative formula to arbitrary $k$0 for square sequence graphs built purely by edge gluing: $k$1. The proof observes that the edge-gluing argument never uses $k$2 specifically, only the inductive hypothesis at level $k$3.

Proof architecture

The induction proceeds by analyzing the union $k$4 through the intersection $k$5. In the edge-gluing case, the intersection is contractible (it is an embedded join over an edge), so Lemma on contraction gives $k$6, and $k$7 reduces to the standard join $k$8 by the sphere-join lemma.

The corner-gluing case is substantially more involved. There, $k$9 is a cone $k=2$0 with apex $k=2$1, and the intersection $k=2$2 is not contractible in itself but is contractible in $k=2$3 (since $k=2$4 is a wedge of 3-spheres and $k=2$5 is 2-dimensional). The contraction-suspension lemma then yields $k=2$6. A careful analysis shows $k=2$7 — argued either via Quillen's fiber theorem applied to a simplicial map from the suspension, or via a homotopy pushout argument — and that $k=2$8 is obtained from $k=2$9 by filling exactly those cycles lying in $k$0, giving

$k$1

Two suspensions then land in dimension 3, matching the edge-glued contribution. This dimensional bookkeeping — each attachment contributing precisely one new 3-sphere per existing square — is what makes the binomial count work uniformly across attachment types.

Application to total-$k$2-cut complexes

The paper's second motivation comes from total cut complexes $k$3, introduced in prior work generalizing Fröberg's theorem and Eagon–Reiner's results relating chordality, clique complexes, and 2-linear resolutions. The key structural observation is that $k$4: the two complexes are combinatorial Alexander duals, so their stable homotopy types are linked.

Applying Alexander duality to the main theorem yields a generalization of earlier results for narrow grids ($k$5 for all $k$6, and $k$7 for $k$8) to arbitrary grids:

$k$9

Since Alexander duality determines only (co)homology, the author must additionally verify simple connectivity of $G_{m,n}$0: for $G_{m,n}$1, every triple of vertices fails to be an independent set complement obstruction, so the complex contains all 2-simplices and every cycle is filled. The homology Whitehead theorem then upgrades the homological equivalence to a genuine homotopy equivalence. Notably, working with the low-dimensional robust clique complex avoids the heavier combinatorial machinery used in prior treatments of the high-dimensional cut complexes directly.

Limitations and open questions

The paper is explicit about where its methods stop. The enumerative formula of the main theorem does not extend to large $G_{m,n}$2: for $G_{m,n}$3, there are exactly two disjoint maximum independent sets of size 6, and the complex is $G_{m,n}$4 rather than a wedge of $G_{m,n}$5 nine-spheres. When $G_{m,n}$6, direct computation always applies, giving a point (if $G_{m,n}$7 is odd) or a single sphere (if $G_{m,n}$8 even), reflecting the one-or-two structure of maximum independent sets in grids.

The author conjectures that for all $G_{m,n}$9, the complex of any square sequence graph is still a wedge of $k \in \{2,3\}$0-spheres with count given by a recurrence, and suspects the closed binomial formula holds when $k \in \{2,3\}$1 is small relative to $k \in \{2,3\}$2. The stated technical bottleneck is the recursive step for $k \in \{2,3\}$3 under corner gluing, where the current argument relies on $k \in \{2,3\}$4 in two places: the dimension of the inductive wedge and the contractibility-in-$k \in \{2,3\}$5 of the 2-dimensional intersection. The corner-gluing analysis also leaves one step — verification that the suspension map induces a homotopy equivalence via Quillen's theorem — deferred to the reader. Finally, the degenerate case $k \in \{2,3\}$6 or $k \in \{2,3\}$7 (paths) is excluded from the main theorem and handled separately via known results on cut complexes of paths plus duality.

Conclusion

The paper establishes clean wedge-of-spheres homotopy types for $k \in \{2,3\}$8-robust clique complexes of grid-like graphs at levels $k \in \{2,3\}$9, with an explicit binomial count tied to the number of squares, and transfers these results to total cut complexes through Alexander duality combined with a simple connectivity check. Its technical core — a König-theoretic decomposition coupled with the embedded join and a homotopy pushout analysis of corner attachments — provides a template that plausibly extends to higher $2k-3$0, though the failure of the formula near the independence number shows the regime of validity is genuinely bounded. The main open question is whether the recurrence-based wedge description holds for all $2k-3$1 and whether the binomial formula survives for $2k-3$2 small relative to $2k-3$3.