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Classification of polyhedral graphs by numbers of common neighbours

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Pith's one-line read The paper proves a complete three-way classification of polyhedra by their sets of common-neighbour counts.

desk verdict A credible and substantial classification of polyhedral graphs by common-neighbour sets, with one unshipped finite enumeration worth verifying before acceptance. read the letter →

arxiv 2508.01349 v1 pith:NXIUCNVI submitted 2025-08-02 math.CO

Riccardo W. Maffucci This is my paper · ORCID
classification math.CO MSC 05C1005C7505C6905C1205E3052B0505C85
keywords planargraphpolyhedroncommonneighbours3-polytopeDezagraphsstronglyregularradiusclassification
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

This paper proposes a complete classification of polyhedra—planar, 3-connected graphs—by their type, defined as the set of numbers of common neighbours that occur among pairs of distinct vertices. For every finite set $A$ of non-negative integers, the author claims exactly one of three outcomes: all polyhedra of type $A$ are explicitly classified, infinitely many are constructed, or none exist. The proof splits at whether $1\in A$: when $1\notin A$ every polyhedron lies in the short family $S_1$ (bipyramids, the graphs $T_\ell$, and ten exceptional graphs, Table 1), and when $1\in A$ the type must appear in Table 2, where the remaining infinite families are pyramids, $W_3$, $W_4$, $B_\ell/B'_\ell$, or the explicitly constructed unions with $A'_n$. This gives a complete answer to a question that generalizes the earlier classification of regular polyhedra by common-neighbour types.

What carries the argument

The argument is carried by the type set $A(G)=\{|N(u,v)| : u,v\in V(G), u\ne v\}$ together with three structural tools. First, the reduction lemmas show that in a planar graph $2\in A$ whenever some $a\ge3$ lies in $A$, and $2\in A$ is equivalent to containing a 4-cycle, so the absence of $2$ forces $A\subseteq\{0,1\}$. Second, for radius-1 polyhedra the plane neighbourhood $\Gamma_u(G)$ of a dominating vertex is a pyramid, so the graph is a pyramid plus added edges; this yields the characterizations of $W_3$ and $W_4$. Third, for radius at least 2 and $0\notin A$, a bound $p\le4M^2+3M+2$ on the order follows from domination in planar graphs of diameter 2, which is what turns the types $\{1,2\},\{1,2,3\},\{1,2,4\}$ into finite searches plus explicit families.

What would settle it

Re-run the enumeration of Section 2 with an independent, certified program that lists all 3-connected planar triangulations on at most 12 vertices and checks whether $1\notin A(G)$; finding any such graph beyond the six named $S_2,S_3,S_4,S_8,S_9,S_{10}$ (and beyond bipyramids and $T_\ell$) would disprove Theorem 1.3.

Watch

Extended reading notes

Core claim

The central discovery is that the common-neighbour spectrum $A(G)$ is a classifying invariant for polyhedra: the paper proves a trichotomy for every finite set $A$ of non-negative integers. If $1\notin A$, Theorem 1.3 states that $G\in S_1$ exactly, with the full type decomposition in Table 1—tetrahedron, cube, icosahedron, octahedron, the graphs $T_3, T_4, T_\ell$, the ten exceptions $S_1,\dots,S_{10}$, and the bipyramids. If $1\in A$, Theorem 1.4 asserts that $A(G)$ is one of the listed types, and for each listed type the corresponding polyhedra are either characterized (e.g. $A=\{0,1\}$ exactly when no 4-cycles are present), bounded in order by $24,47,78$ outside the named families, equal to $B_\ell$ or $B'_\ell$ for even $\ell\ge6$, or proven to occur infinitely often via explicit constructions. Together these two theorems exhaust all possible types.

Load-bearing premise

The completeness of Table 1 rests on a computer enumeration of all triangulations with up to 12 vertices, reported in Section 2; if that enumeration missed any triangulation with $1\notin A$, the classification would be incomplete and later statements relying on Table 1 would inherit the gap.

Editorial extensions

If this is right

  • For every finite set $A$ of non-negative integers, exactly one of the trichotomy outcomes holds: all polyhedra of type $A$ are classified, infinitely many exist, or none exist.
  • A polyhedron has $A=\{0,1\}$ if and only if it contains no 4-cycles, hence every polyhedron of girth at least 5 has type $\{0,1\}$.
  • All polyhedra of type $\{1,2\}$, $\{1,2,3\}$, or $\{1,2,4\}$ are, outside orders $p\le24$, $p\le47$, and $p\le78$ respectively, exactly the $n$-gonal pyramids, the class $W_3$, and the class $W_4$.
  • For even $\ell\ge6$, the only polyhedra with $1\in A$, $0,3,4\notin A$, and $\ell\in A$ are $B_\ell$ and $B'_\ell$.
  • A polyhedron on at least 25 vertices has $A=\{0,1,2\}$ if and only if it is not a pyramid, contains a 4-cycle, and contains no subgraph isomorphic to $K(2,3)$.

Reading between the lines

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

  • The trichotomy is special to planarity: Lemma 3.1 does not hold for general graphs, since complete graphs satisfy $A(K_n)=\{n-2\}$, so the same three-outcome classification cannot be expected for all graphs.
  • The order bounds $24,47,78$ come from a generic domination bound for planar diameter-2 graphs; a sharper bound of that kind would shrink the finite checks and could make the classification effectively verifiable by enumeration.
  • The caterpillar and gluing constructions of Section 6 provide infinite families with prescribed common-neighbour sets; these families could serve as explicit generators for testing network algorithms that use common-neighbour similarity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper classifies polyhedra (planar 3-connected graphs) by their type A(G), the set of all numbers |N(u,v)| of common neighbours over pairs of distinct vertices. Theorem 1.3 characterizes all polyhedra with 1 not in A: they are exactly the family S1 (bipyramids, T_l = P_l + K2, and ten exceptional graphs), with the type list of Table 1. Theorem 1.4 covers every polyhedron with 1 in A through Table 2: for the types {1,2}, {1,2,3} and {1,2,4}, everything outside a finite order bound (p <= 24, 47, 78) lies in a named infinite family (pyramids, W3, W4); the type {1,2,l} with l even >= 6 is exactly B_l or B'_l; the type {0,1} is exactly the class of polyhedra with no 4-cycles; {0,1,2} is characterized for order at least 25; and three construction propositions in Section 6 yield infinitely many polyhedra for the types {1,2,3} union A'_4, {1,2,4} union A'_5, and {0,1,2} union A'_3. Section 5 proves that no other types occur, giving the abstract's trichotomy: for every finite set A, the authors classify all polyhedra of that type, or construct infinitely many, or prove that none exist. The strategy combines a diameter-two domination bound (Lemma 3.5, via Goddard and Henning), a classification of radius-one polyhedra (Lemmas 4.1-4.3 and Propositions 4.4-4.5), a structural theorem for the type {1,2,l} (Proposition 5.1), and a small computer enumeration in Section 2 for triangulations of maximum degree at most 5.

Significance. If correct, the trichotomy is a complete, sharp classification of all planar 3-connected graphs by their common-neighbour spectrum, substantially extending the author's earlier classification of planar Deza graphs [8]; the statement is falsifiable, since a single polyhedron whose type is absent from Tables 1 and 2 would refute it. The analytic core is genuinely parameter-free: I re-derived the degree-bound reduction p <= 12 in Section 2 for Delta <= 5, the bound p <= 4M^2 + 3M + 2 in Lemma 3.5 from the Goddard-Henning domination input, the 24/47/78 arithmetic in Corollary 4.6, and the case split of Proposition 5.2, and found them consistent. The constructions in Section 6 are explicit, and Propositions 6.1 and 6.2 ship executable Mathematica verification code, a genuine reproducibility asset. Two caveats are load-bearing: the completeness of Theorem 1.3 (and the contrapositive step G not in S1 implies 1 in A that feeds Proposition 4.4 and hence most of Table 2) rests on an unshipped computer enumeration, and the proof of Lemma 3.6 contains an invalid counting step, although the lemma's statement is true.

major comments (2)
  1. [Section 2 (enumeration of triangulations with up to 12 vertices)] Section 2, paragraph following the equation 12 = 3p3 + 2p4 + p5 >= p: the only-if direction of Theorem 1.3 for triangulations of maximum degree at most 5 is completed solely by the sentence stating that all triangulations with up to 12 vertices were inspected, with code available on request. This step is load-bearing: a missed triangulation (for instance one with type {0,2} or {2,4} on at most 12 vertices) would falsify Table 1, and the contrapositive that G not in S1 implies 1 in A is used in Proposition 4.4, hence in Proposition 4.5 and Corollaries 4.6 and 4.7, and therefore in Table 2's rows {1,2}, {1,2,3}, {1,2,4}, and {0,1,2}. The code is not shipped: Appendix A provides Mathematica code only for Propositions 6.1 and 6.2, so this step is neither derived nor certified in the manuscript. Since there are only about 9,000 unlabeled triangulations of the sphere with up to 12 vertices, the authors should ship the generating and checking code together with a certificate (for instance the complete list of such triangulations and their type sets), or replace the enumeration with a hand-checkable argument.
  2. [Section 3, Lemma 3.6 and Eq. (3.1)] Section 3, Lemma 3.6, Eqs. (3.1) and the preceding paragraph: the deduction that f3 = f/2 does not follow from the statement that, for each vertex, exactly half of the faces containing it are triangular. Summing that statement over all vertices gives 3f3 = sum_{i>=5} i f_i (there are no 4-cycles, since A = {1} contradicts Corollary 3.3), which implies only f3 >= 5f/8; the displayed chain 2q >= 3f/2 + 5 sum_{i>=5} f_i = 3f/2 + 5(f - f3) = 3f/2 + 5f/2 = 4f uses f3 = f/2 in the last equality, so the contradiction 4p + 4r - 8 = 4q >= 4p + 4r is not obtained as written. The preceding pairing argument (u, v1, v2, then u, v3, v4, and so forth) also assumes without proof that the perfect matching induced by unique common neighbours pairs consecutive vertices in the cyclic order around u. The lemma's statement is nevertheless true: with A = {1}, every pair of vertices has exactly one common neighbour, and the Friendship Theorem forces a windmill graph, which is not 3-connected when it has at least two triangles (and K3 is not a polyhedron), so Corollary 3.7 and Proposition 5.2 stand once the proof is repaired. The proof as printed needs that repair.
minor comments (4)
  1. [Section 3, Lemma 3.5] The proof of Lemma 3.5 excludes the unique graph of Figure 9 from the Goddard-Henning domination statement [4, Theorem 2], but does not verify the claimed bound p <= 4M^2 + 3M + 2 for that exceptional graph. Since the applications in Corollary 4.6 require only the bounds 24, 47, and 78, an explicit check of that single graph, or a sentence explaining that its order lies below these bounds, would close the gap.
  2. [Section 5, Proposition 5.2] In the first case of Proposition 5.2, the proof that 2 not in A implies A = {0,1} rules out A = {1} via Lemma 3.6, but does not rule out A = {0} or A = empty; these are impossible for a connected graph of order at least 4 with diameter at least 2, and one sentence would suffice.
  3. [Section 1, definition of S1] The symbol S1 is overloaded: it denotes both the family S1 := {bipyramids} union {T_l} union {S_i} and the exceptional graph S1 (the cube) in Figure 2 and Table 1; this should be disambiguated to avoid confusion in the statements of Theorem 1.3 and Proposition 4.4.
  4. [Sections 1 and 6, typography] The phrase 'Theorems 1.3 and Theorems 1.4' in Section 1 should read 'Theorems 1.3 and 1.4', and the opening sentence of Section 6, 'To complete the proof of Theorem 1.3', should refer to Theorem 1.4, since Theorem 1.3 is proved in Section 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivations are self-contained; the only external step is an unshipped finite enumeration, which is a reproducibility concern, not circularity.

full rationale

The paper's main claims (Theorems 1.3 and 1.4) are proved by structural arguments on plane graphs. Theorem 1.3 reduces the 1-not-in-A case to three regimes: quadrangulation (cube), triangulation with maximum degree at least 6 (bipyramids, T_l, S5-S7), and triangulations with maximum degree at most 5. In the last regime, the paper invokes an exhaustive computer check of all triangulations on at most 12 vertices to list six further exceptions. This enumeration is asserted rather than shipped, so it is a verification and completeness risk, but it is not circular: the finite check is independent of the theorem being proved and is not a fitted parameter, a redefinition, or the same as the target classification. The remaining chain—Lemma 3.1 through Corollary 3.7, Propositions 4.4, 4.5, 4.8, 5.1, 5.2, and the constructions of Section 6—does not assume the target classifications. Self-citations to [8], [9], and [10] supply standard or previously established facts (Hamiltonian cycles in 2-connected outerplanar graphs, caterpillar degree sequences, and previously constructed examples for A = {0,1,2}); these are not used in a way that reduces a predicted quantity to its own input. No equation in the paper is equivalent to its assumptions by construction, and no fitted parameter is renamed as a prediction. Therefore no circular step is present, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The classification rests on standard planar graph theory (Euler, Whitney), two external theorems (outerplanar Hamiltonicity; Goddard-Henning domination), and one unshipped computational artifact (the ≤ 12-vertex triangulation enumeration). No free parameters are fitted to data; the construction parameters (pyramid sizes, the sets A′n, ℓ) are inputs to the constructions, not fitted quantities.

assumptions (5)
  • standard math Every 2-connected outerplanar graph is Hamiltonian and contains at least two vertices of degree 2
    Invoked as [9, Lemma 3.3] in Lemma 4.2 (radius-1 polyhedra have pyramid plane neighbourhoods) and in Lemma 4.8 (endblocks of outerplanar graphs have degree-2 vertices). Classical result, proved by the author in [9].
  • domain assumption Every planar graph of diameter 2 has domination number at most 2, except one explicitly listed graph
    Invoked as [4, Theorem 2] (Goddard-Henning) in Lemma 3.5 to derive p ≤ 4M²+3M+2, which produces the classification bounds p ≤ 24/47/78 in Corollary 4.6. Published external theorem, not proved in this paper.
  • domain assumption The computer enumeration of all triangulations on up to 12 vertices is exhaustive and correct
    The completeness of Theorem 1.3 and Table 1 rests on this check ('We inspected all triangulations with up to 12 vertices (code available on request)', end of Section 2). The six exceptional graphs S2, S3, S4, S8, S9, S10 are identified only by this enumeration.
  • standard math Whitney's theorem: every 3-connected planar graph has a unique spherical embedding
    Cited as [16] and used from the introduction on to justify fixing a planar embedding of each polyhedron and the plane-neighbourhood arguments based on (1.2).
  • standard math Euler's formula and the handshaking lemma for planar graphs
    Used throughout, including the degree count in Section 2 (3p3 + 4p4 + 5p5 = 12) and the face count in Lemma 3.6.

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Pith. "Pith review of Classification of polyhedral graphs by numbers of common neighbours." pith.science (2026). https://pith.science/paper/NXIUCNVI

@misc{pith2026250801349,
  author       = {Pith},
  title        = {Pith review of: Classification of polyhedral graphs by numbers of common neighbours},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXIUCNVI}},
  note         = {Machine review of arXiv:2508.01349}
}
abstract

We propose a classification of polyhedra (planar, $3$-connected graphs) according to their type i.e., their set of quantities of common neighbours for each pair of distinct vertices. For every (finite) set of non-negative integers, we either classify all the polyhedra of that type, or construct infinitely many polyhedra of that type, or prove that none exist. This problem is related to the theory of strongly regular and Deza graphs, distances in graphs, and degree sequences. There is potential for application to complex networks and data science.

Figures

Figures reproduced from arXiv: 2508.01349 by the authors.

Figure 1
Figure 1. Tℓ. We define the family of polyhedra S1 := {bipyramids} ∪ {Tℓ : ℓ ≥ 2} ∪ {Si : 1 ≤ i ≤ 10}, where S1, S2, . . . , S10 are the graphs in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The ten exceptional graphs. Definition 1.1. We call W3 the class of polyhedra obtained from an n-gonal pyramid (i.e., wheel graph), n ≥ 5, by adding a certain number of pairwise independent edges (at least one). We call W4 the class of polyhedra G obtained from a pyramid of base [v1, v2, . . . , vn], n ≥ 5 by adding pairwise disjoint cycles (at least one) such that all of the following hold. Each cycle is of length … view at source ↗
Figure 3
Figure 3. A member of W3. Definition 1.2. Let n ≥ 4 be even. We define the graph Bn where we start with the cycle b1, b2, . . . , b2n and add the edges xbi for 1 ≤ i ≤ 2n and ybj j ≡ 1, 2 (mod 4), 1 ≤ j ≤ 2n. We also define the graph B′ n where we start with the path b1, b2, . . . , b2n−2 and add the edges xbi for 1 ≤ i ≤ 2n − 2 and ybj j ≡ 1, 2 (mod 4), 1 ≤ j ≤ 2n − 2. Illustrations may be found in [PITH_FULL_IMAGE:figures/… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Illustrations of B6 and B′ 6 . We are ready to state our main results. Theorem 1.3. A polyhedron G satisfies 1 ̸∈ A if and only if G ∈ S1. We have the classification of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Impossible situations for 1 ̸∈ A. Now by planarity, w1u3, w1u4, w3u1, w3u2 ̸∈ E(G), thus N(w1, w3) = {v}, contradiction. Next, assume by contradiction that G contains adjacent faces [u1, u2, u3, u4] and [u1, u2, v]. Since 1 ̸∈ A, there exists w ∈ V (G), w ̸= v, adjacen…
Figure 6
Figure 6. Figure 6: Case where G is a quadrangulation and 1 ̸∈ A. Henceforth we may assume that G is a triangulation. Let u ∈ V (G) be of maximal degree in G, d = deg(u) = ∆(G), and v1, v2, . . . , vd (2.1) be its neighbours in cyclic order around u in the planar immersion of G. Since G i…
Figure 7
Figure 7. Figure 7: There exists j such that vj is adjacent to all of (2.2). adjacent to v1, v2, . . . , vi−1, vi+1, vi+2, vi+4, vi+5, . . . , vd. Since |N(vi−1, vi+3)| ̸= 1 and by planarity, vi+3vi ∈ E(G) so that the claim is proven. Now assume instead that for all 1 ≤ i < i′ ≤ d with 2 …
Figure 8
Figure 8. Figure 8: Lemma 3.1. Take a subgraph of G isomorphic to the square pyramid, with apex u and base [v1, v2, v3, v4], as in Figure 8b. Now u, v1 have a common neighbour y ̸∈ {v2, v4}, that we may draw w.l.o.g. inside of the cycle u, v1, v2. Next, y, v4 have a common neighbour other…
Figure 9
Figure 9. Figure 9: The only planar graph with diameter 2 and domination number 3. We write the disjoint union V (G) = {x, y} ∪ N(x, y) ∪ Vx ∪ Vy, where Vx is the set of vertices ̸= x, y adjacent to x but not y, and Vy the set of vertices ̸= x, y adjacent to y but not x. As rad(G) ̸= 1, t…
Figure 3
Figure 3. Figure 3: Since G is obtained from H by adding independent edges, we deduce that N(vi) = {u, vi−1, vi+1, vj}. By planarity, vi+1vi−1 ̸∈ E(G). If j ̸= i + 2, again since G is obtained from H by adding independent edges, we have vi+1vj ̸∈ E(G). It follows that N(vi , vi+1) = {u}, …
Figure 10
Figure 10. Figure 10: If G ∈ W4, then 3 ̸∈ A. In fact, we are ready to classify all polyhedra such that A = {1, 2, ℓ} for 2 ≤ ℓ ≤ 4. Corollary 4.6. If G is a polyhedron satisfying A = {1, 2}, then either G is a pyramid or p ≤ 24. If G is a polyhedron satisfying A = {1, 2, 3}, then either G…
Figure 11
Figure 11. Figure 11: Proposition 5.1. Along P, for every i ≥ 1 the vertices wi , wi+1, wi+2 cannot be consecutive, otherwise N(wi , wi+2) ⊇ {u, v, wi+1} thus by planarity N(wi , wi+2) = {u, v, wi+1}, impossible. Hence along P either w1, u, w2, w3, v, w4, . . . , wℓ appear in this order, o…
Figure 12
Figure 12. Figure 12: Proposition 6.1. For A′ 4 = {6, 7, 10, 12}, we consider the caterpillar with central vertices of degrees 4, 5, 8, 10. If A′ 4 = {8}, we take the star with central vertex of degree 6. A possible code for Proposition 6.1, in the Mathematica language, may be found in App…
Figure 13
Figure 13. Figure 13: Proposition 6.2: the graph G − u for A′ 5 = {5, 8}. The central vertices of the caterpillar have degrees 3, 3, 6. The 3 is repeated since A′ 5 contains an odd number of odd elements [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Gi,j . Writing A ′ 3 = {a1, a2, . . . , am}, we take the graphs G1,a1 , G2,a2 , . . . , Gm,am. (6.1) For 1 ≤ i ≤ m − 1, we identify (i.e, ‘glue’) the region wi,ai , xi,ai , yi,ai , zi,ai , of Gi,ai with the region ui+1,ai+1 , bi+1,ai+1 , vi+1,ai+1 , ci+1,ai+1 , of Gi+…
Figure 15
Figure 15. Figure 15: Code for Proposition 6.1, in the Mathematica language. [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: Code for Proposition 6.2, in the Mathematica language. [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]

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