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A Weight Function Lemma Heuristic for Graph Pebbling

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By building Weight Function Lemma certificates from shortest paths to the farthest vertices first, this paper improves the best-known pebbling upper bounds for the Blanuša snark and the Flower snarks.

desk verdict Solid WFL lower-bound theorem plus concrete improved bounds for snarks; the main gap is under-specified certificates in the appendix, which should be fixable before publication. read the letter →

arxiv 2505.16050 v1 pith:KHOHKDQP submitted 2025-05-21 cs.DM

G. A. Bridi , F. L. Marquezino , C. M. H. de Figueiredo This is my paper · ORCID
classification cs.DM MSC 05C3590C2705C85
keywords GraphpebblingWeightFunctionLemmaSnarksHeuristicsWFLratioDualformulationnumberBlanuša
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

Graph pebbling asks how many pebbles guarantee that any chosen target can be reached, and computing the answer is hard enough that the practical route is bounding. This paper works inside the Weight Function Lemma, the standard tool for upper bounds, but instead of solving the usual integer linear program it uses the dual problem, where a certificate is a hand-buildable set of weighted subtrees. The paper proves a new lower bound on the best WFL ratio any such certificate can achieve, showing that the farthest vertices from the target force surplus weight onto closer neighborhoods. From that diagnosis it designs a heuristic: build one subtree per neighbor of the target, route weight along shortest paths to the farthest vertices first, then fill the remaining weights. Applied to snark families, the heuristic improves the best-known upper bounds, giving $\pi(B_2)\le 30$ (previously 34) for the Blanuša snark and $\pi(J_m)\le \lfloor 2^{k+2}\cdot 8/5+2k-8/5\rfloor+1$ for Flower snarks $J_m$ with $m=2k+1\ge5$, and a companion bound $\pi(B_1)\le31$.

What carries the argument

The carrying object is a WFL certificate: a set of rooted subtrees, one per neighbor of the target, each with a weight function in which a parent's weight is at least twice its child's, so that the total weight divided by the minimum weight over non-root vertices bounds the pebbling number from above. The paper adds two pieces of machinery: Theorem 2.1, which lower-bounds that ratio by counting the surplus that peripheral vertices force onto small neighborhoods, and the trunk-and-branch heuristic, which assigns trunk weights $2^{e(r)-1-d_{T_j}(v_j,v)}$ along shortest paths from neighbor $v_j$ to every peripheral vertex and then adds branches until every vertex reaches the common minimum weight $\omega_{\min}$ from Eq. (5). The WFL ratio of the assembled certificate is what Tables 1 and 2 compare against previous bounds.

What would settle it

Recompute the distance table of Appendix D for a concrete Flower snark, say $J_7$: check $d_{J_7}(z_0,\alpha_j)=j+1$ for $\alpha\in\{v,x,y\}$ and $1\le j\le3$, and check that each listed trunk is a shortest path in $J_7-\{z_0\}$. If any distance is off by one, the trunk weight $2^{e(z_0)-1-d}$ assigned in Figure 7 changes and some parent-child inequality $\omega_T(\mathrm{parent})\ge 2\omega_T(\mathrm{child})$ fails, so the computed ratio $\lambda_{J_7}=55.6$ and the bound $\pi(J_7)\le56$ are invalid; a correct recalculation that confirms all distances and inequalities would settle the certificate.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the dual WFL certificate problem has a structural bottleneck: the target's farthest (peripheral) vertices are what set the minimum weight, and every shortest path from a neighbor of the target to those peripheral vertices forces surplus weight on intermediate neighborhoods when those neighborhoods are small. Theorem 2.1 quantifies this as $\lambda_{G,r}\ge n(G)-1+\sum_{j\in I_{\mathrm{sur}}}(2^{e(r)-j}-|N_j(r)|)$, where $I_{\mathrm{sur}}$ collects the neighborhoods too small to absorb the required weight. The heuristic is the constructive counterpart: it chooses the minimum weight via Eq. (5), builds each subtree's trunk from shortest paths to peripheral vertices, and adds branches only to lift underweighted vertices up to the minimum. The paper reports that this construction yields valid WFL certificates for the Blanuša and Flower snarks, improving the previously known upper bounds and matching the new lower bound exactly for some targets, such as $z_0$ in $J_3$, where the WFL ratio reaches 12.

Load-bearing premise

The load-bearing premise is that the distance values and shortest-path uniqueness asserted for the Flower snarks in Appendices D and E are correct for every $m$; in particular $d_{J_m}(z_0,\alpha_j)=j+1$ for $\alpha\in\{v,x,y\}$, and if any one of those distances or uniqueness claims fails, the trunk weights and branch placements no longer satisfy the WFL parent-child condition and Theorem 3.2 collapses.

Editorial extensions

If this is right

  • If the bounds are correct, a 30-pebble configuration on $B_2$ is always solvable to any target, improving the old guarantee of 34 pebbles.
  • For every Flower snark $J_m$ with $m=2k+1\ge5$, the pebbling number is at most $\lfloor 2^{k+2}\cdot 8/5+2k-8/5\rfloor+1$; in particular the paper gives $\pi(J_5)\le29$ and $\pi(J_7)\le56$.
  • The lower bound in Theorem 2.1 shows that the dual WFL method cannot certify anything below $n(G)-1+\sum_{j\in I_{\mathrm{sur}}}(2^{e(r)-j}-|N_j(r)|)$ for a target $r$, so a certificate reaching that value is optimal within this method; the paper exhibits such a certificate for $z_0$ in $J_3$.
  • The heuristic narrows the gap between lower and upper bounds for the two snark families, with the asymptotic gap for Flower snarks reduced from the old factor $9/5$ to $8/5$ times the dominant $2^{k+2}$ term.
  • The companion appendix extends the same certificate construction to the first Blanuša snark, giving $\pi(B_1)\le31$.

Reading between the lines

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

  • Beyond the paper, the same trunk-first construction is a general recipe for any target whose only bottleneck neighborhood is the first one ($I_{\mathrm{sur}}=\{1\}$): Eq. (5) supplies the starting weights, and the remaining work is a filling problem that could be automated for larger graph families.
  • A consequence the authors leave implicit is that whenever the heuristic reaches the Theorem 2.1 bound, the certificate is provably optimal among all dual WFL certificates, so a tighter pebbling upper bound for that target would require a method outside the WFL dual formulation.
  • Testable extension: a machine-checked verification of the distance tables and shortest-path uniqueness for all $m$ would convert the hand-built infinite-family certificate in Appendix E into a formally checked proof, eliminating the main verification burden.
  • Because the tight case of Theorem 2.1 requires every neighbor of the target to lie on a shortest path to every peripheral vertex, highly geodesic graphs are natural testbeds; the hypercube example in Appendix C suggests the condition is satisfiable in a range of symmetric graphs.
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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

3 major / 3 minor

Summary. The paper studies the Weight Function Lemma (WFL) dual formulation for graph pebbling bounds. It proves a new lower bound on the optimal dual WFL ratio λ_{G,r}, namely λ_{G,r} ≥ n(G)−1 + Σ_{j∈I_sur}(2^{e(r)−j} − |N_j(r)|) (Theorem 2.1), and proposes a heuristic that builds WFL certificates by pairing each neighbor of the target with a strategy whose trunk consists of shortest paths to all peripheral vertices. The heuristic is applied to the Blanuša graph B2 and to Flower snarks J_m, yielding the claimed upper bounds π(B2)≤30 (Theorem 3.1) and π(J_m)≤⌊2^{k+2}·8/5 + 2k − 8/5⌋+1 for m=2k+1≥5 (Theorem 3.2). These would improve the corresponding bounds of Adauto et al. for most target vertices.

Significance. If the certificates are correct, the paper makes a solid contribution: Theorem 2.1 is a genuinely new lower bound on the best WFL ratio, and the heuristic produces concrete improvements over the state of the art on an infinite family of snarks. The proof of Theorem 2.1 is clean, and the arithmetic in Tables 1–5 is internally consistent in the spot checks I made (e.g., 58.5/2=29.25 for B2 target x1, and the J5/J7 values 28 and 55.6 for the v0 row). The paper clearly explains the heuristic and compares against prior bounds at target-level granularity, which is valuable. However, as written, the main new upper bounds rest on certificates whose tree structure is not fully specified for several targets, and on distance and shortest-path uniqueness claims in Appendix D that are asserted rather than proved. These are verifiability gaps in the load-bearing part of the argument rather than observed contradictions.

major comments (3)
  1. [Appendix D and Section 3 (Flower snarks)] The distance formulas and uniqueness claims that drive the heuristic are stated without proof. For example, d_{J_m}(z0,α_j)=j+1 for α∈{v,x,y}, d_{J_m}(x0,y_i)=i+2, and the assertion that the shortest path in G−{z0} between each neighbor of z0 and each z0-peripheral vertex is unique are all listed as facts. These values determine the trunk weights through Eq. (5) and the size of the neighborhoods used in Appendix D, and they feed directly into the certificate weights in Appendix E. A single incorrect distance or a non-unique shortest path would break the WFL parent-child inequalities and with them Theorem 3.2. Please provide proofs of these distance/uniqueness statements, or a small verification script.
  2. [Appendix E (certificates for targets x0 and v0 of Flower graphs)] The certificates for x0 and v0 are presented only as vertex lists with weight tuples; no parent–child edge sets are given. Under Lemma 1.1, a strategy is a tree and must satisfy ω_parent ≥ 2·ω_child on every internal edge, so a flat list of vertices and weights is not an independently checkable certificate. This is not a cosmetic omission: the v0 certificate is the bottleneck in Table 2 and determines the graph bound in Theorem 3.2. Please exhibit the tree structure explicitly (e.g., parent lists, adjacency in the figures, or a machine-readable certificate file).
  3. [Tables 3 and 5 (B1 and B2 certificates)] The same verifiability issue appears in the B1/B2 certificates. The text says that trunk vertices are highlighted in bold, but the tables as typeset contain no bold, and no parent–child directions are shown. Since Theorem 3.1 and the B1 bound in Appendix F rest on these tables, the WFL halving condition cannot be checked from the printed material. Please add explicit parent lists or edge information for each strategy row.
minor comments (3)
  1. [Theorem 3.2] The theorem is typeset as ⌊2k+28/5 + 2k−8/5⌋+1, which reads as a linear term in k. It should be ⌊2^{k+2}·8/5 + 2k − 8/5⌋+1, consistent with Table 2 and Appendix E.
  2. [Theorem 3.1] The proof says there are six vertex classes represented by targets x1, x1, x3, z1, z2, z3; the second x1 should presumably be x2, and the target classes should be listed without duplication.
  3. [Appendix C] In the cube-graph tight-bound construction, the inequality a_j < (d−j+1)a_{j−1} is asserted with only the condition d−j+1<d−1; an explicit justification or a reference for this inequality would help the reader follow the proof.

Circularity Check

1 steps flagged · score 2.0 of 10

Core results align with inputs only through explicit, checkable structures: lower bounds derive from the WFL halving condition and upper bounds from displayed certificates. The single minor self-citation is the baseline comparison with the authors' own prior paper, not a load-bearing reduction; no fitted parameter is renamed as a prediction.

  1. other [Section 1 (Introduction) and Section 3 (Tables 1–2, comparison with Ref. [5])]
    "Their work is a pioneering study on the pebbling number of snark graphs, and by focusing on the same graph families, we provide a direct and fair comparison between our heuristic results and the state of the art. ... Theorem 3.1 improves the first in 4 units, while Theorem 3.2 improves the last one by a constant factor of 9/8."

    The 'state of the art' baseline for every improvement claim is Ref. [5], which shares co-author C. M. H. de Figueiredo with the present paper, so the novelty framing is measured against a self-cited prior result. This is a minor self-reference, not a load-bearing circularity: Theorems 3.1 and 3.2 are established by explicit r-strategy certificates whose weights are displayed in Figures 2–7 and Appendices D–E and satisfy the WFL parent-child condition, and Theorem 2.1 is proved from Eq. (1) plus that same condition. No equation assumes the target bound, and no fitted parameter is renamed as a prediction; the reported λ ratios are computed from constructed weights.

full rationale

The derivation chain is self-contained. Theorem 2.1 is a first-principles lower bound: from λ_T = n−1 + Σ(ω_T(v)−ω_min)/ω_min (Eq. 1), the proof bounds each neighborhood's surplus using only the WFL halving condition along paths from r-peripheral vertices; it assumes no target result. The heuristic (Section 3) uses Theorem 2.1's structure (Eq. 5 sets ω_min from graph distances), but the upper bounds of Theorems 3.1 and 3.2 are computed from the displayed certificates: each strategy is a subtree rooted at r with weights satisfying ω_parent ≥ 2ω_child, and λ_T = |ω_T|/ω_min is evaluated arithmetically (Appendix E), so WFL yields π ≤ ⌊λ_T⌋+1. These ratios are genuinely smaller than [5]'s for most targets (λ_{B2} ≤ 29.25 vs 236/7; J_m z0 coefficient 3/2 vs 9/5), and the authors explicitly acknowledge the v0 certificate matches [5]'s v0 ratio. The self-reference to [5] (shared authorship) frames the contribution but does not enter the proofs; the certificates are independently checkable. Two weaknesses are recorded as non-circular: (i) Appendix E gives x0/v0 strategies as flat vertex/weight lists without explicit parent-child edges, and Appendix D asserts distance and shortest-path-uniqueness data without proof, so independent verification of Theorem 3.2 requires reconstructing those trees — a completeness/correctness risk, not a circular step; (ii) Theorem 3.2's displayed formula loses the exponent in '2^{k+2}·8/5', confirmed by Table 2 and the Appendix E computation. The Final Remarks candidly state the heuristic leaves a gap to the lower bound and that steps 3–4 are manual, consistent with a non-circular but heuristic method.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted: all weights are derived from the WFL condition and the graph geometry. The paper introduces no new entities. The main unproved inputs are the WFL lemma itself (cited) and the graph-specific distance data in Appendix D.

assumptions (3)
  • standard math Weight Function Lemma (Lemma 1.1) is assumed from Hurlbert [3], converting weight certificates into pebbling upper bounds.
    The paper's entire bounding framework rests on this cited lemma; it is not reproved.
  • domain assumption The distance formulas and shortest-path uniqueness for Jm and B2 in Appendix D are assumed without proof.
    The general Flower-snark certificate in Appendix E uses these distances to set trunk weights via Eq. (5); an error here would invalidate Theorem 3.2.
  • standard math The dual reduction to a single normalized parameter (1/alpha = alpha_T for all T) is treated as without loss of generality.
    Section 1 after Lemma 1.1; rescaling each strategy's weight function absorbs the alpha_T coefficients, which is valid but stated rather than shown in detail.

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Pith. "Pith review of A Weight Function Lemma Heuristic for Graph Pebbling." pith.science (2026). https://pith.science/paper/KHOHKDQP

@misc{pith2026250516050,
  author       = {Pith},
  title        = {Pith review of: A Weight Function Lemma Heuristic for Graph Pebbling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KHOHKDQP}},
  note         = {Machine review of arXiv:2505.16050}
}
read the original abstract

Graph pebbling is a problem in which pebbles are distributed across the vertices of a graph and moved according to a specific rule: two pebbles are removed from a vertex to place one on an adjacent vertex. The goal is to determine the minimum number of pebbles required to ensure that any target vertex can be reached, known as the pebbling number. Computing the pebbling number lies beyond NP in the polynomial hierarchy, leading to bounding methods. One of the most prominent techniques for upper bounds is the Weight Function Lemma (WFL), which relies on costly integer linear optimization. To mitigate this cost, an alternative approach is to consider the dual formulation of the problem, which allows solutions to be constructed by hand through the selection of strategies given by subtrees with associated weight functions. To improve the bounds, the weights should be distributed as uniformly as possible among the vertices, balancing their individual contribution. However, despite its simplicity, this approach lacks a formal framework. To fill this gap, we introduce a novel heuristic method that refines the selection of balanced strategies. The method is motivated by our theoretical analysis of the limitations of the dual approach, in which we prove lower bounds on the best bounds achievable. Our theoretical analysis shows that the bottleneck lies in the farthest vertices from the target, forcing surplus weight onto the closer neighborhoods. To minimize surplus weight beyond the theoretical minimum, our proposed heuristic prioritizes weight assignment to the farthest vertices, building the subtrees starting from the shortest paths to them and then filling in the weights for the remaining vertices. Applying our heuristic to Flower snarks and Blanu\v{s}a snarks, we improve the best-known upper bounds, demonstrating the effectiveness of a structured strategy selection when using the WFL.

Figures

Figures reproduced from arXiv: 2505.16050 by the authors.

Figure 1
Figure 1. Adapted representation from Ref. [3] of the optimal [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Application of the heuristic method to the target [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Application of the heuristic method to the target [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Flower graphs (a) J3 and (b) J5. B Complete set of z0-strategies on Flower graphs In Figures 5, 6, and 7, we present visual representations for all z0-strategies T = {T1, T2, T3} of the Flower graphs J3, J5, and Jm (m = 2k + 1, m > 5), respectively. These figures compl…
Figure 5
Figure 5. Figure 5: Set T = {T1, T2, T3} of z0-strategies for the Flower graph J3, where (a) shows T1, (b) T2, and (c) T3. We omit the root z0 in the representation of the subtrees, and we represent the trunk edges and branches by double lines and standard lines, respectively. 12 [PITH_F…
Figure 6
Figure 6. Figure 6: Set T = {T1, T2, T3} of z0-strategies for the Flower graph J5, where (a) shows T1, (b) T2, and (c) T3. We omit the root z0 in the representation of the subtrees, and we represent the trunk edges and branches by double lines and standard lines, respectively. 2 v0 k+1 v−…
Figure 7
Figure 7. Figure 7: Set T = {T1, T2, T3} of z0-strategies for the Flower graph Jm, with m = 2k + 1, m > 5, where (a) shows T1, (b) T2, and (c) T3. We omit the root z0 in the representation of the subtrees, and we represent the trunk edges and branches by double lines and standard lines, r…
Figure 8
Figure 8. Figure 8: Blanuˇsa graph B1. r T Subtrees Weights N(r) P(r) |ωT | ωmin λT a1 T1 (a ′ 1 , b2, b ′ 2 , c2, c ′ 2 , d1, d ′ 1 , e1, e2, e ′ 1 , e ′ 2 ) (8, 4, 4, 2, 2, 1, 1, 1/2, 1/2, 1/2, 1/2) {a ′ 1 , b1, b′ 1 } {e1, e2, e′ 1 , e′ 2 T2 (b1, c1, d2, e1, e2, d1, d } 60 2 30 ′ 1 , e…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The only Class 0 Flower snark is the smallest

    math.CO 2025-05 conditional novelty 6.0 of 10

    J_3, the smallest Flower snark, is Class 0 with pebbling number 12, making it the only Class 0 Flower snark.

Reference graph

Works this paper leans on

39 extracted references · 35 canonical work pages · cited by 1 Pith paper

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    (8,4,5/2,2,5/4,1,1,1/2) {z1, x3, x4} {x′ 1, x′ 2, x′ 5, z′ 1} 58.5 2 29.25T2 (x3,z 3, x5,x ′ 3,x ′ 1,x ′ 5,x ′ 2,z ′ 1, z′

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    (8,4,2,2,1,1,1/2,1/2,1/4) T3 (x4,z 4, x2,x ′ 4,x ′ 1,x ′ 2,z ′ 1, z′

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    (8,4,2,2,1,1,1/2,1/2) x2 T1 (z2,z ′ 2, z1,x ′ 2,z ′ 1,x ′ 1,x ′ 5,x ′ 3, x′ 4, z′

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    (8,4,5/2,2,2,1,1,1/2,1/2,1/2) {z2, x5, x4} {x′ 4, x′ 5, z′ 2} 66.5 2.5 26.6T2 (x5,x 3,z 5,z 3,z ′ 5,x ′ 3,x ′ 5,x ′ 1, x′

  11. [19]

    (8,4,4,2,2,1,1,1/2,1/2) T3 (x4,z 4, x1,x ′ 4,z 3,x ′ 1,x ′ 3,x ′ 5, z′

  12. [20]

    (8,4,5/2,2,2,1,1,1/2,1/2) x3 T1 (z3,z 4,x ′ 3,x ′ 4,x ′ 1,x ′ 2, x′ 5,z ′ 1,z ′

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    (8,4,5/2,2,5/4,1,1,1/2,1/2) {z3, x1, x5} {x′ 2, z′ 1, z′ 2} 58.5 2 29.25T2 (x1,z 1, x4,z 2,z ′ 2,x ′ 2,z ′ 1, x′

  14. [22]

    (8,4,2,2,1,1/2,1/2,1/4) T3 (x5,z 5, x2,z ′ 5,x ′ 5,z ′ 1, x′ 1,x ′ 2,z ′

  15. [23]

    (8,4,2,2,1,1,1/2,1/2,1/2) z1 T1 (x1,x 3,x 4,z 3,z 4,x ′ 3,x ′ 4,x ′ 1, x′ 2, x′

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    (8,4,4,2,2,1,1,1/2,1/2,1/2) {x1, z2, z5} {x′ 1, x′ 3, x′ 4} 66.5 2.5 26.6T2 (z2,z ′ 2, x2,x ′ 2,z ′ 1,x ′ 1,x ′ 4,x ′ 3, z4) (8,4,5/2,2,2,1,1,1/2,1/2) T3 (z5,z ′ 5, x5,x ′ 5,z ′ 1,x ′ 1,x ′ 3,x ′ 4, z3) (8,4,5/2,2,2,1,1,1/2,1/2) z2 T1 (z′ 2,z ′ 1,x ′ 1, x′ 2,x ′ 3, x′ 4, x′ 5,...

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    (8,4,2,2,1,1,1,1/2) T3 (z1,z 5,x 5,z ′ 5,x 3,x ′ 5,x ′ 3,z 3) (8,4,2,2,1,1,1/2,1/2) z3 T1 (z4,x 4,x ′ 4,x 2,x ′ 2,z 2,z ′ 2, z1, z′

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    (8,4,4,2,2,1,1,1/2,1/2) {z4, x3, x′ 3} {z2, z′ 2} 68 2.5 27.2T2 (x3,x 1, x5,z 1, z5,z 2, x2,z ′ 2, z′

  19. [27]

    (8,4,4,2,2,1,1/2,1/2,1/2) T3 (x′ 3,x ′ 1, x′ 5,z ′ 1, z′ 5,z ′ 2, x′ 2,z 2, z5) (8,4,4,2,2,1,1/2,1/2,1/2) Table 3: SetTofr-strategiesT 1,T 2, andT 3 for all targetsrof the Blanuˇ sa graphB 2. r T Subtrees W eights N(r) P(r) |ωT | ωmin λT x0 T1 (z0,v 0, y0,v 1,v −1, z1, z−1) (4...

  20. [28]

    (8,4,4,2,2,1,1,1/2,1/2,1/2,1/2) {a′ 1, b1, b′ 1} {e1, e2, e′ 1, e′ 2} 60 2 30T2 (b1,c 1,d 2,e 1,e 2, d1, d′ 1,e ′ 1,e ′

  21. [29]

    (8,4,2,1,1,1/2,1/2,1/2,1/2) T3 (b′ 1,c ′ 1,d ′ 2,e ′ 1,e ′ 2, d1, d′ 1,e 1,e 2) (8,4,2,1,1,1/2,1/2,1/2,1/2) b1 T1 (a1,b ′ 1, a′ 1,c 2,c ′ 1,d 1,d ′ 2,e ′

  22. [30]

    (8,4,5/2,2,2,1,1,1/2) {a1, c1, c′ 2} {d1, d′ 2, e′ 2} 65 2.5 26T2 (c1,d 2, b2,e 1,e 2,d 1,e ′ 2, c2,d ′ 2, e′

  23. [31]

    (8,4,5/2,2,2,1,1,1/2,1/2,1/2) T3 (c′ 2,d ′ 1, b′ 2,e 2,e ′ 1,d ′ 2,e ′ 2, c′ 1,d 1, e1) (8,4,5/2,2,2,1,1,1/2,1/2,1/2) c1 T1 (b1,c ′ 2, a1,b ′ 2, d′ 1,c ′ 1,d ′

  24. [32]

    (8,4,2,2,2,1,1/2) {b1, b2, d2} {c′ 1, d′ 2} 58 2 29T2 (b2,c 2, a′ 1,b ′ 1,d 1,c ′ 1,e ′ 2,d ′

  25. [33]

    (8,4,2,2,2,1,1,1/2) T3 (d2,e 1, e2,e ′ 1,d ′ 2, e′

  26. [34]

    (8,4,2,2,1,1) d1 T1 (c2,b 2, b′ 1,a ′ 1,c 1, a1,b 1,b ′ 2, c′ 1,c ′

  27. [35]

    (8,4,7/2,2,2,7/4,1,1,1/2,1/2) {c2, e1, e′ 2} {b1, b′ 2, c′ 2} 70.5 2.5 28.2T2 (e1,d 2,e ′ 1,c 1,d ′ 1,b 1,c ′ 2, a1,b ′

  28. [36]

    (8,4,4,2,2,1,1,1/2,1/2) T3 (e′ 2,d ′ 2,e 2,c ′ 1,d ′ 1,b ′ 2,c ′ 2, a′ 1,b 1, a1) (8,4,4,2,2,1,1,1/2,1/2,1/4) e1 T1 (d1,c 2,b 2,b ′ 1,a 1,a ′ 1,b ′

  29. [37]

    (8,4,2,2,1,1,1/2) {d1, d2, e′ 1} {a1, a′ 1, b′ 2} 58 2 29T2 (d2,c 1,b 1, e2,a 1,c ′ 2,a ′ 1,b ′

  30. [38]

    (8,4,2,2,1,1,1/2,1/2) T3 (e′ 1,d ′ 2,c ′ 1, d′ 1, e′ 2,b ′ 2, c′ 2,a ′

  31. [39]

    For each strategy, we omit the root and highlight the vertices of the trunk in bold

    (8,4,2,2,2,1,1,1/2) Table 5: SetTofr-strategiesT 1,T 2, andT 3 for all targetsrof the Blanuˇ sa graphB 1. For each strategy, we omit the root and highlight the vertices of the trunk in bold. 18

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.