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Asymmetric induced saturation

T0 review · 2 major / 1 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read H-deletion-saturated graphs exist for every non-complete graph on at most six vertices and for several infinite families of larger graphs.

desk verdict The paper proves existence of H-deletion-saturated graphs for several families and all H with at most 6 vertices using explicit constructions. read the letter →

arxiv 2606.24763 v1 pith:42PJNF3Y submitted 2026-06-23 math.CO

Xinyue Fan , Sahab Hajebi , Sepehr Hajebi , Sophie Spirkl This is my paper · ORCID
classification math.CO
keywords inducedsaturationdeletion-saturatedgraphsH-inducedsubgraphsgraphtheoryconjecturesbipartitelineoftreesunicyclic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper defines an H-deletion-saturated graph as one that contains at least one edge, contains no induced copy of a fixed graph H, yet produces an induced copy of H upon deletion of any single edge. It conjectures that such graphs exist for every non-complete H and proves the claim for complete bipartite graphs with unequal part sizes, triangle-free graphs containing exactly one cycle, graphs possessing two leaves at distance at most three, and line graphs of trees, with the arguments covering substantially more general families inside each class. The authors also verify the conjecture by direct checking for every possible H on at most six vertices. The deletion-only version is presented as more tractable than the symmetric version that requires both addition and deletion of edges to create induced copies of H.

What carries the argument

An H-deletion-saturated graph: an edge-containing graph free of induced H such that every single-edge deletion produces an induced copy of H.

What would settle it

A counterexample would be any non-complete graph H on six or fewer vertices together with a demonstration that no H-deletion-saturated graph exists for it.

Watch

Extended reading notes

Core claim

The authors establish that for every graph H on at most six vertices there exists a graph G with at least one edge containing no induced copy of H such that the deletion of any edge of G produces an induced copy of H. They further construct such graphs G for all complete bipartite graphs H with unequal part sizes, all triangle-free unicyclic graphs H, all graphs H with two leaves at distance at most three, and all line graphs of trees H, with the constructions extending to more general families within each category.

Load-bearing premise

The structural features of the considered families of H are sufficient to permit explicit construction of one graph G that avoids induced H while every edge deletion forces an induced H.

Editorial extensions

If this is right

  • Such an H-deletion-saturated graph exists for any complete bipartite H with parts of different sizes.
  • Such a graph exists for any triangle-free H that contains exactly one cycle.
  • Such a graph exists for any H that has exactly two leaves at distance at most three.
  • Such a graph exists for any H that is the line graph of a tree.
  • The existence holds for every H with at most six vertices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The results indicate that completeness of H may be the only obstruction to existence of deletion-saturated graphs.
  • Constructions for these families could be adapted to test the conjecture on other specific infinite families of graphs.
  • If the conjecture holds in general it would separate the difficulty of the asymmetric deletion version from the symmetric add-and-delete version of induced saturation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript defines H-deletion-saturated graphs G (containing at least one edge, no induced copy of a fixed H, but every single edge deletion creates an induced H) and conjectures that such G exists for every non-complete H. It proves the conjecture for complete bipartite graphs with unequal part sizes, triangle-free unicyclic graphs, graphs with two leaves at distance at most three, line graphs of trees (and substantially more general families in each case), as well as all H on at most six vertices, via explicit constructions and exhaustive small-case verification.

Significance. If the constructions hold, the work supplies concrete supporting evidence for the conjecture by resolving it for multiple infinite families and all small-order H. The emphasis on explicit constructions (rather than non-constructive existence arguments) is a methodological strength, as it permits direct checking of the two required properties.

major comments (2)
  1. [Abstract] Abstract and introduction: the central existence claims for the four listed families rest on explicit constructions asserted to satisfy both the no-induced-H property and the every-edge-deletion property, yet the manuscript supplies neither proof sketches nor an outline of the case analysis used to verify these properties. Without these details the load-bearing step from construction to verified saturation cannot be inspected.
  2. [Introduction] The claim that the constructions extend to 'substantially more general families' is stated without a precise definition of the enlarged families or an indication of which additional structural hypotheses are relaxed while preserving the saturation property.
minor comments (1)
  1. Notation for the distance condition on leaves and for the line-graph case should be introduced with a short example to avoid ambiguity in the general-family statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. We address each major comment below and will revise the manuscript accordingly to improve clarity on the verification of constructions and the definitions of generalized families.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the central existence claims for the four listed families rest on explicit constructions asserted to satisfy both the no-induced-H property and the every-edge-deletion property, yet the manuscript supplies neither proof sketches nor an outline of the case analysis used to verify these properties. Without these details the load-bearing step from construction to verified saturation cannot be inspected.

    Authors: We agree that the abstract and introduction would benefit from additional guidance on the verification process. In the revised manuscript we will insert concise proof sketches and outlines of the case analyses (with pointers to the relevant sections) for each of the four families, making the transition from construction to verified saturation properties explicit and inspectable. revision: yes

  2. Referee: [Introduction] The claim that the constructions extend to 'substantially more general families' is stated without a precise definition of the enlarged families or an indication of which additional structural hypotheses are relaxed while preserving the saturation property.

    Authors: The body of the paper defines the enlarged families and the relaxed hypotheses in the respective sections. To address the concern, we will add a clarifying paragraph in the introduction that explicitly names each generalized family, states the additional structural hypotheses that are relaxed, and cross-references the precise definitions and theorems where the saturation property is proved for those families. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation to prior result on even cycles; all new existence claims rest on explicit constructions

full rationale

The paper's central results consist of explicit combinatorial constructions proving existence of H-deletion-saturated graphs for the listed families (unequal bipartite, unicyclic triangle-free, two leaves at distance ≤3, line graphs of trees, and all |V(H)|≤6). These constructions are presented as direct and independent of any fitted parameters or self-referential definitions. The only self-citation is the reference to the authors' recent proof for even cycles, which is not load-bearing for the new families or the conjecture verification on small H. No step reduces by construction to its inputs, no ansatz is smuggled, and no uniqueness theorem is invoked from self-work. The derivation chain is therefore self-contained against external combinatorial verification.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work uses only the standard definitions and axioms of simple undirected graphs; no numerical parameters are fitted, no new entities are postulated, and no ad-hoc axioms beyond classical graph theory are invoked.

assumptions (1)
  • standard math Graphs are finite, simple, undirected, with no loops or multiple edges.
    Invoked throughout the definition of induced subgraphs and edge deletion.

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Pith. "Pith review of Asymmetric induced saturation." pith.science (2026). https://pith.science/paper/42PJNF3Y

@misc{pith2026260624763,
  author       = {Pith},
  title        = {Pith review of: Asymmetric induced saturation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42PJNF3Y}},
  note         = {Machine review of arXiv:2606.24763}
}
abstract

For which graphs $H$ does there exist a graph $G$ with at least one edge and no induced subgraph isomorphic to $H$, such that deleting any edge of $G$ creates an induced copy of $H$? We call such a graph "$H$-deletion-saturated". This version of the well-studied notion of "$H$-induced-saturated" graphs -- where both adding and deleting any edge creates an induced copy of $H$ -- appears more tractable. For example, while it remains wide open whether $H$-induced-saturated graphs exist for every even cycle $H$, we proved recently that deletion-saturated graphs exist for all even cycles. In fact, apart from complete graphs, no graph $H$ is known for which $H$-deletion-saturated graphs do not exist. We conjecture that $H$-deletion-saturated graphs exist for every non-complete graph $H$, and prove this conjecture for several types of graphs, including: complete bipartite graphs with parts of unequal size, triangle-free graphs with one cycle, graphs with two leaves at distance at most three, and line graphs of trees. In fact, in all cases, we prove the conjecture for substantially more general families. We also verify our conjecture for every graph $H$ on at most six vertices.

Figures

Figures reproduced from arXiv: 2606.24763 by the authors.

Figure 1
Figure 1. The icosahedron. (In fact, they prove a stronger result, namely that there is a countably infinite H-free graph G where any “bounded-degree perturbation” of G creates an induced copy of H.) Our focus is on graphs H for which there exists a finite H-induced-saturated graph. We call such graphs normal. (From here on, all graphs in this paper have finite vertex sets, no loops, and no parallel edges.) It follows that co… view at source ↗
Figure 2
Figure 2. Proof of Theorem 3.3. Recall that the icosahedron is C4-induced-saturated [3]. Since K2,2 is the 4-cycle, each of W1, . . . , Ws is K2,2-induced-saturated, and in particular K2,2-free. Now, • For each i ∈ {1, . . . , s}, fix an edge uivi ∈ E(Wi). • For every w ∈ V (G) \ {z1, . . . , zs ′+2}, say w ∈ V (Wi) for some i ∈ {1, . . . , s}, fix an edge uwvw of Wi such that uww, vww /∈ E(G). Note that the choice of uwvw in… view at source ↗
Figure 3
Figure 3. Left: A graph H with a governing block B. Right: All blocks of H except B are omnipresent in the icosahedron. Consequently, D, D′ 1 , . . . , D′ s are pairwise anticomplete in L; in particular, D, D′ 1 , . . . , D′ s are pairwise disjoint. Since x, y ∈ D, we deduce that D, D′ 1 , . . . , D′ s are pairwise anticomplete in L + xy. In conclusion, we have shown that D, D ′ 1 , . . . , D ′ s are s+1 pairwise anticomplete… view at source ↗
Figures from the paper (28 more)
Figure 4
Figure 4. Figure 4: Proof of (15). The hyperedge φ(y) and sets Di,ℓ (left) and the new hyperedge f (right). D1,l D2,l C2,l φ(y2) φ(x1) φ(x2) D1,l f1 φ(x1) D1,l φ(y1) C1,l D2,l φ(x2) D2,l f2 C2,l C1,l [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Proof of (16). The hyperedges φ(y1), φ(y2) and sets Di,ℓ (left) and the new hyperedges f1 and f2 (right). Thus, φ(x1) and φ(x2) are two disjoint ℓ-pendent hyperedges of Γℓ . In addition, it is straightforward to check that, since φ is an isomorphism between H and L(Γ),…
Figure 6
Figure 6. Figure 6: Proof of (17). The hyperedges φ(y1), φ(y2) and sets Di,ℓ (left) and the new hyperedges f1 and f2 (right). (17) If α(NH(y1)) + α(NH(y2)) ≥ d + 2, then for every ℓ ∈ {1, . . . , d − 1}, there is a d-graph Γℓ with |V (Γℓ)| < n + ℓ and two disjoint ℓ-pendent hyperedges suc…
Figure 7
Figure 7. Figure 7: Induced subgraph obstructions to line graphs [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Induced subgraph obstructions to biline graphs. A.3. Near-line-graphs and severed pairs. We say that a graph H is a biline graph biline graph if there exists a bipartite graph F such that H = L(F). There are well-known characterizations of both line graphs and biline g…
Figure 9
Figure 9. Figure 9: Proof of Theorem A.5. Graphs H1, H2, H3, H4, along with the complement of H3 and H4 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: Proof of Theorem A.6. The 6-cycle, the complement of the 6-cycle, and the 5-cycles. Proof. Let H1 be entry 3F in [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Proof of Theorem A.7. From left to right, the graphs H1, . . . , H5 (top) and their complements (bottom). Proof. Note that entry 5F in [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: Proof of Theorem A.8: The graph H1 and its complement, two drawings of H2, two drawings of H3, the graph H4, and the graph G. • e = u1u11. Then, for S = {u1, u3, u6, u7, u8, u11}, it is readily seen that the map f : V (H1) → S with f(v1) = u6, f(v2) = u1, f(v3) = u11,…
Figure 13
Figure 13. Figure 13: Proof of Theorem A.9: Two drawings of the graph H (left) and the graph G (right) [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]
Figure 14
Figure 14. Figure 14: Proof of Theorem A.10. From left to right: Two drawings of the graph H, and the graph G. This proves (19). The result is now immediate from (18) and (19). This completes the proof of Theorem A.9. ■ At long last, we are now left with the 8 graphs in Group 3, for which …
Figure 15
Figure 15. Figure 15: Proof of Theorem A.11. From left to right: Graphs H, G1 and G. It remains to show that for every e ∈ E(G), the graph G − e has an induced subgraph isomorphic to H. Since G is edge-transitive, it suffices to prove the latter statement for only one edge e ∈ E(G). Let e …
Figure 16
Figure 16. Figure 16: Proof of Theorem A.12. From left to right: Graphs H, G1 and G, and the subgraph of G induced by the neighborhood of u1. Our goal is to show that G is H-deletion-saturated. First, we show that G is H-free. Suppose for a contradiction that G has an induced subgraph H′ i…
Figure 17
Figure 17. Figure 17: Proof of Theorem A.13. From left to right: The graph H and its complement, and the graph G with a depiction of E(G1), E(G2), and M. obtained from the (disjoint) union of G1 and G2 by adding the matching M = {u 1 i u 2 i : i ∈ {1, . . . , 10}}. See [PITH_FULL_IMAGE:fi…
Figure 18
Figure 18. Figure 18: Proof of Theorem A.14. The graph H (left) and its complement (right). in G1 from u 1 i1 to u 1 i2 . Thus, C = u 1 i1 -u 1 i2 -u 1 i3 -u 1 i4 -u 1 i1 is an induced 4-cycle in G + e. Let NG1 (u 1 i3 ) \ {u 1 i2 , u1 i4 } = {u 1 i5 }, where i5 ∈ {1, . . . , 10} \ {i1, i2…
Figure 19
Figure 19. Figure 19: Proof of Theorem A.15. The graph H (left) and the Schläfli graph G (right). Our goal is now to prove that the Schläfli graph G is H-deletion-saturated. To see that G is H-free, suppose for a contradiction that G has an induced subgraph isomorphic to H. It follows in p…
Figure 20
Figure 20. Figure 20: Proof of Theorem A.16. From left to right: The graph H and its complement, the graph G and its construction. (22) G is 2P3-free. Suppose for a contradiction that there are two anticomplete induced copies P 1 and P 2 of P3 in G. Let I1 = V (G) \ N[V (P 1 )] and let I2 …
Figure 21
Figure 21. Figure 21: Proof of Theorem A.17. From left to right: The graph H and its complement, and the graph G. Proof. Let H be entry 15I in [PITH_FULL_IMAGE:figures/full_fig_p038_21.png]
Figure 22
Figure 22. Figure 22: All pairwise non-isomorphic non-complete graphs on at most six vertices [PITH_FULL_IMAGE:figures/full_fig_p039_22.png]
Figure 23
Figure 23. Figure 23: All non-complete complete multipartite graphs on at most six vertices [PITH_FULL_IMAGE:figures/full_fig_p040_23.png]
Figure 24
Figure 24. Figure 24: These graphs have at least one isolated vertex, and removing all isolated vertices from each graph leaves a non-complete graph [PITH_FULL_IMAGE:figures/full_fig_p041_24.png]
Figure 25
Figure 25. Figure 25: These graphs each have a governing block B, and at least one other block, and all blocks other than B are isomorphic to K2 [PITH_FULL_IMAGE:figures/full_fig_p042_25.png]
Figure 26
Figure 26. Figure 26: These graphs are all line−graphs. In each entry, we have a drawing of the graph H itself with a specified non-edge e, a drawing of an induced subgraph H′ of H that by Theorem A.2 prevents H from being a line graph, and then a drawing of the graph F whose line graph is…
Figure 27
Figure 27. Figure 27: These graphs are all line−graphs. In each entry, we have a drawing of the graph H itself with a specified non-edge e, a drawing of an induced subgraph H′ of H that by Theorem A.2 prevents H from being a line graph, and then a drawing of the graph F whose line graph is…
Figure 28
Figure 28. Figure 28: These graphs are all complements of line+graphs. In each entry, we have a drawing of the graph H itself, a drawing of H with a specified edge e of H, a drawing of an induced subgraph H′ of H that by Theorem A.2 prevents H from being a line graph, and then a drawing of…
Figure 29
Figure 29. Figure 29: These graphs are all biline−graphs. In each entry, we have a drawing of the graph H itself with a specified non-edge e, a drawing of an induced subgraph H′ of H that by Theorem A.3 prevents H from being a biline graph, and then a drawing of the bipartite graph F whose…
Figure 30
Figure 30. Figure 30: These graphs are all complements of biline+graphs. In each entry, we have a drawing of the graph H itself, a drawing of H with a specified edge e of H, a drawing of an induced subgraph H′ of H that by Theorem A.3 prevents H from being a biline graph, and then a drawin…
Figure 31
Figure 31. Figure 31: Every graph here is the line graph of a graph F with a severed pair. In each entry, we have a drawing of the graph H itself, and then a drawing of the graph F with a severed pair (x1, x2) such that L(F) = H [PITH_FULL_IMAGE:figures/full_fig_p047_31.png]

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    •e=u 1 1u2

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    This proves (23)

    In this case,P1 =u 1 6-u1 1-u2 4 andP 2 =u 1 3-u2 3-u2 5 work. This proves (23). The result now follows from (22) and (23). This completes the proof of Theorem A.16. ■ Theorem A.17.Entry15Iin Figure 22 is deletion-normal. 38 ASYMMETRIC INDUCED SATURATION Figure 21.Proof of The...

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