4-cycle-free induced subgraphs of grid graphs
Pith reviewed 2026-05-10 16:45 UTC · model grok-4.3
The pith
Maximal vertex sets inducing C4-free subgraphs in 2D grid graphs are explicitly characterized.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors characterize the maximal sets S of vertices in a 2-dimensional grid graph G such that the induced subgraph on S is free of 4-cycles. They additionally establish upper and lower bounds on the number of C4-free induced subgraphs whose vertex sets have size slightly smaller than this maximum.
What carries the argument
The explicit structural characterization of maximal C4-free induced subgraphs on the rectangular lattice vertices under standard grid adjacency.
If this is right
- The maximum number of vertices selectable while keeping the induced subgraph C4-free is now known for every finite grid.
- The number of such near-maximal selections lies between two explicit functions of the grid dimensions.
- Any algorithm that enumerates or samples C4-free induced subgraphs on grids can now be bounded in the near-maximal regime.
Where Pith is reading between the lines
- The same structural description may help classify maximal induced subgraphs avoiding other small cycles or forbidden patterns on the same grids.
- The counting bounds could be used to estimate the complexity of search problems that require large cycle-free induced subgraphs.
- Periodic or toroidal boundary conditions on grids might admit similar characterizations with only local adjustments.
Load-bearing premise
The characterization and bounds apply specifically to finite rectangular 2D grids with ordinary horizontal and vertical edges and with induced subgraphs taken in the strict vertex-subset sense.
What would settle it
A single counterexample would be any vertex set larger than the described maximum that still induces no 4-cycle, or any enumeration of near-maximal C4-free induced subgraphs whose count falls outside the stated upper and lower bounds.
read the original abstract
The avoidance of induced forests, or induced acyclic subgraphs, in $d$-dimensional grid graphs, or lattice graphs, has been studied in Alon et al. and later in Caragiannis et al., finding upper and lower bounds with respect to the number of vertices in a single dimension $n$ and the dimension $d$. In this work, we study the avoidance of induced $C_4$-free subgraphs, a superset of induced forests, of $2$-dimensional grid graphs $G$ and characterize the maximal sets $S \subseteq V$ such that the induced subgraph $G_S$ of $G$ with vertex set $S$ is $C_4$-free. Additionally, we will give upper and lower bounds on the number of $C_4$-free induced subgraphs with slightly fewer vertices than contained in the maximum.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper characterizes the inclusion-maximal vertex sets S in finite 2-dimensional grid graphs such that the induced subgraph on S is C4-free, and supplies upper and lower bounds on the number of C4-free induced subgraphs whose order is slightly smaller than the maximum.
Significance. A complete, boundary-aware characterization of maximal C4-free induced subgraphs would usefully extend the existing literature on induced forests in grids to the strictly larger class of C4-free graphs. The accompanying enumeration bounds, if tight, would give concrete quantitative information about the size of the family of near-maximal examples.
major comments (2)
- [Characterization theorem (presumably §3 or §4)] The central characterization theorem must explicitly treat boundary vertices (those of degree less than 4). The abstract and the structural argument appear to rely on interior lattice patterns; without a dedicated case analysis or exhaustive enumeration of boundary configurations, it is possible that additional maximal C4-free sets exist that are not captured by the claimed description. This directly affects the completeness claim.
- [Bounds section (presumably §5)] The upper and lower bounds on the number of near-maximal C4-free induced subgraphs are stated in terms of the grid dimensions n and m. The proof of these bounds should be checked for dependence on the same boundary handling; if the counting argument inherits the same interior-only assumption, the bounds may fail to be tight for small or narrow grids.
minor comments (2)
- [Introduction and notation] Notation for the grid graph G_{n,m} and the induced subgraph G[S] should be introduced once and used consistently; the abstract uses both G_S and G[S] interchangeably.
- [Introduction] The paper cites Alon et al. and Caragiannis et al. for the induced-forest case; a brief comparison table or paragraph clarifying how the C4-free extremal function relates to the forest extremal function would help readers.
Simulated Author's Rebuttal
We thank the referee for their thorough review and valuable suggestions. We address each major comment below and describe the corresponding revisions to the manuscript.
read point-by-point responses
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Referee: [Characterization theorem (presumably §3 or §4)] The central characterization theorem must explicitly treat boundary vertices (those of degree less than 4). The abstract and the structural argument appear to rely on interior lattice patterns; without a dedicated case analysis or exhaustive enumeration of boundary configurations, it is possible that additional maximal C4-free sets exist that are not captured by the claimed description. This directly affects the completeness claim.
Authors: We appreciate the referee highlighting the need for explicit boundary treatment. Our characterization (Theorem 4.1) defines maximal C4-free sets using local forbidden configurations that apply uniformly, automatically restricting possibilities at boundary vertices due to their lower degrees. However, to remove any ambiguity and ensure completeness, we have added a dedicated subsection 4.2 providing exhaustive case analysis for corners, edges, and boundary-interior transitions. We have also revised the abstract to note that the characterization is boundary-aware. revision: yes
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Referee: [Bounds section (presumably §5)] The upper and lower bounds on the number of near-maximal C4-free induced subgraphs are stated in terms of the grid dimensions n and m. The proof of these bounds should be checked for dependence on the same boundary handling; if the counting argument inherits the same interior-only assumption, the bounds may fail to be tight for small or narrow grids.
Authors: The bounds in Section 5 are obtained by enumerating near-maximal extensions of the characterized maximal sets, and the revised characterization now incorporates boundary effects explicitly. The counting formulas include perimeter adjustments for the grid. We have added a remark verifying the bounds hold for small and narrow grids (including explicit checks for n,m ≤ 5) and updated the proofs to display the boundary contributions. revision: yes
Circularity Check
No circularity; characterization rests on direct structural arguments
full rationale
The paper characterizes maximal C4-free induced subgraphs of finite grid graphs via direct analysis of the rectangular lattice adjacency and induced-subgraph definition. No equations, fitted parameters, self-citations, or uniqueness theorems appear in the abstract or described approach. The central claims do not reduce by construction to their own inputs, and the derivation is self-contained against the grid structure without self-referential steps.
discussion (0)
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