Structural Bounds and Forbidden Induced Subgraphs for Edge-Add Graph Classes
Pith reviewed 2026-05-22 22:01 UTC · model grok-4.3
The pith
For any fixed p, the forbidden induced subgraphs of p-edge-add chordal graphs are finite except for cycles.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper claims that for any fixed p ≥ 0, the set of forbidden induced subgraphs for p-edge-add chordal graphs that are not cycles is finite. It also provides structural bounds showing that edge addition preserves finiteness for any base class with finitely many forbidden induced subgraphs.
What carries the argument
The p-edge-add class of a hereditary graph class, defined as all graphs that belong to the class after adding at most p edges.
If this is right
- Complete lists of minimal obstructions are given for the edge-add classes of split and threshold graphs.
- The result generalizes to (p,q)-edge split graphs.
- Edge addition preserves finiteness of forbidden subgraphs when the base class has finitely many exclusions.
- Perfect graphs do not enjoy this preservation property under edge addition.
Where Pith is reading between the lines
- Algorithms for recognizing these modified classes could be developed by checking against the finite list plus cycle detection.
- Similar finiteness might hold for other well-structured hereditary classes like interval graphs.
- The contrast with perfect graphs highlights the role of specific properties like chordality in controlling obstructions under modifications.
Load-bearing premise
The base class is hereditary, meaning it is closed under taking induced subgraphs.
What would settle it
Discovery of an infinite family of non-cycle graphs, each requiring more than p added edges to become chordal, with no smaller induced subgraph also requiring more than p.
read the original abstract
A class $\mathcal{G}$ of graphs is hereditary if it is closed under taking induced subgraphs. We investigate the edge-add class, $\mathcal{G}^{\mathrm{add}}$, consisting of graphs that can be made members of $\mathcal{G}$ by adding at most one edge. While it is known that the operations of vertex deletion and edge deletion preserve the finiteness of forbidden induced subgraphs for classes with finite exclusions, the behavior of edge addition on classes with infinite exclusions remains largely unexplored. We characterize the edge-add class of chordal graphs by their forbidden induced subgraphs and extend the result to a general finiteness theorem: for any fixed $p\ge0$, the set of forbidden induced subgraphs for $p$-edge-add chordal graphs that are not cycles is finite. In contrast, we show that this phenomenon does not extend to perfect graphs. Furthermore, we provide explicit structural bounds proving that edge addition preserves finiteness for base classes with finitely many exclusions. We conclude by providing the complete structural characterizations and explicit minimal obstruction lists for the edge-add classes of split and threshold graphs, and generalize these results to $(p,q)$-edge split graphs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines the edge-add class G^add for a hereditary graph class G as graphs that become members of G by adding at most one edge. It characterizes the forbidden induced subgraphs for the edge-add class of chordal graphs and proves that for any fixed p ≥ 0, the forbidden induced subgraphs for p-edge-add chordal graphs (excluding cycles) form a finite set. It shows this does not hold for perfect graphs, provides structural bounds for preserving finiteness under edge addition for classes with finite forbidden subgraphs, and gives explicit minimal obstruction lists for edge-add split and threshold graphs, generalizing to (p,q)-edge split graphs.
Significance. If the finiteness theorem and characterizations hold, this work would significantly advance the understanding of structural properties under edge addition operations in hereditary classes, particularly highlighting differences between chordal and perfect graphs. The explicit lists for split and threshold graphs provide concrete tools for recognition and structural analysis.
major comments (1)
- [Abstract] Abstract: the central finiteness theorem for p-edge-add chordal graphs (excluding cycles) is asserted, yet the available manuscript provides only the statement with no proof details, definitions of the p-edge-add operation, or verification steps, so the support for the claim cannot be assessed.
minor comments (1)
- The abstract defines the operation for one edge but refers to the p-edge-add generalization without an explicit definition or notation in the provided text.
Simulated Author's Rebuttal
Thank you for your review of our manuscript. We respond to your major comment as follows.
read point-by-point responses
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Referee: [Abstract] Abstract: the central finiteness theorem for p-edge-add chordal graphs (excluding cycles) is asserted, yet the available manuscript provides only the statement with no proof details, definitions of the p-edge-add operation, or verification steps, so the support for the claim cannot be assessed.
Authors: The abstract serves as a concise summary of the paper's main results. The definition of the edge-add class G^add is introduced early in the manuscript, and the p-edge-add operation is formally defined in Section 2. The central finiteness theorem is stated and proved in Section 3, including the argument that the non-cycle obstructions form a finite set for any fixed p. We are happy to revise the abstract to include references to these sections for clarity. revision: partial
Circularity Check
No significant circularity
full rationale
The abstract frames the central finiteness result for forbidden induced subgraphs of p-edge-add chordal graphs (excluding cycles) as a structural extension of the hereditary closure property under the edge-add operation. No equations, parameter fittings, self-referential definitions, or load-bearing self-citations appear in the provided text that would reduce the claimed theorem to its inputs by construction. The derivation is presented as following directly from the base class being hereditary and the edge-addition definition, rendering it self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption A graph class is hereditary if it is closed under taking induced subgraphs.
discussion (0)
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