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Recovering Conductances of Resistor Networks in a Punctured Disk

Analysis · math.CO · math.SP · posed by Yulia Alexandr, Brian Burks, Sunita Chepuri, Patricia Commins · open

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Statement

A graph is irreducible by Y−ΔY-\Delta moves, pendant removal, self-edge removal, parallel reductions, series reductions, antenna jumping, and antenna absorption if and only if it has three medial strands which pairwise intersect twice, there is a medial strand that self-intersects, and it is irreducible by Y−ΔY-\Delta moves, pendant removal, self-edge removal, parallel reductions, series reductions,and antenna jumping.

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  • Recovering Conductances of Resistor Networks in a Punctured Disk
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed Conjecture 2: for a connected resistor network in a punctured disk Γ\Gamma, with medial graph M(Γ)M(\Gamma), Γ\Gamma is irreducible by

    Y ⁣− ⁣Δ, pendant removal, self-edge removal, parallel/series reductions, antenna jumping, antenna absorptionY\!-\!\Delta,\ \text{pendant removal, self-edge removal, parallel/series reductions, antenna jumping, antenna absorption}

    iff Γ\Gamma is irreducible by the same moves except antenna absorption, M(Γ)M(\Gamma) has three medial strands pairwise intersecting twice, and some medial strand self-intersects.

    Result: The conjecture is false.

    Take Γ\Gamma to be the graph K4K_4 embedded as a wheel: three boundary vertices v1,v2,v3v_1,v_2,v_3 on the outer boundary forming a triangle, and the unique interior boundary vertex bb joined to all three viv_i. Thus

    E(Γ)={v1v2,v2v3,v3v1,bv1,bv2,bv3}.E(\Gamma)=\{v_1v_2,v_2v_3,v_3v_1,bv_1,bv_2,bv_3\}.

    This is a nontrivial rnpd: bb is surrounded by the triangle.

    All four vertices are boundary vertices, so the response matrix is just the weighted graph Laplacian. Its six off-diagonal entries are exactly the six edge conductances. Hence the response set contains a 66-dimensional open set. Any graph with fewer than six edges has a response map depending rationally on at most five conductance parameters, so its image has Zariski dimension at most 55, and cannot contain that open set. Therefore Γ\Gamma is not electrically equivalent to any graph with fewer edges. In particular it is irreducible by any sequence of the listed local moves, including antenna absorption.

    Now compute its medial strands. Let

    ei=vivi+1,fi=bvie_i=v_iv_{i+1},\qquad f_i=bv_i

    with indices mod 33. Tracing straight through the medial vertices gives the three strands

    S1:me1,mf2,mf3,me3,S2:me1,mf1,mf3,me2,S3:me2,mf2,mf1,me3.\begin{aligned} S_1&: m_{e_1},m_{f_2},m_{f_3},m_{e_3},\\ S_2&: m_{e_1},m_{f_1},m_{f_3},m_{e_2},\\ S_3&: m_{e_2},m_{f_2},m_{f_1},m_{e_3}. \end{aligned}

    Thus each pair of strands intersects exactly twice:

    S1∩S2={me1,mf3},S1∩S3={mf2,me3},S2∩S3={mf1,me2}.S_1\cap S_2=\{m_{e_1},m_{f_3}\},\quad S_1\cap S_3=\{m_{f_2},m_{e_3}\},\quad S_2\cap S_3=\{m_{f_1},m_{e_2}\}.

    But no strand self-intersects: each SiS_i passes through four distinct medial vertices, and every medial vertex is shared by two distinct strands.

    So Γ\Gamma satisfies the left-hand side of the conjecture but fails the required self-intersecting-strand condition. Hence Conjecture 2, as stated, is false.

    Audit: the counterexample is connected, planar, has exactly one interior boundary vertex, is nontrivial, and irreducibility was proved in the stronger electrical sense, so no extra assumptions are being used.

    Citation: Conjecture and terminology: Yulia Alexandr, Brian Burks, Sunita Chepuri, Patricia Commins, “Recovering Conductances of Resistor Networks in a Punctured Disk,” arXiv:1812.01517, Conjecture 2. Counterexample above.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The proposed K4K_4 wheel is a valid nontrivial rnpd with three outer boundary vertices and the interior boundary vertex. Since all vertices are boundary vertices, its response matrix is exactly the weighted Laplacian, with six independent off-diagonal conductances. Thus its response set contains a 6-dimensional open set, while any graph with fewer than six edges has response image of dimension at most five; hence it cannot be electrically equivalent to any fewer-edge graph. This implies irreducibility under any sequence of the listed response-preserving moves.

    The medial strand computation is also consistent: the three strands pairwise share exactly two medial vertices, and no strand repeats a medial vertex, so there is no self-intersecting strand. Therefore the graph satisfies the conjecture’s left-hand side but fails the stated right-hand condition. I found no existing literature explicitly giving this counterexample/resolution.

    Novelty assessment

    TYPE1

    Classification rationale: The counterexample is a very small observation: the K4K_4 wheel is elementary, and its irreducibility/medial-strand check is routine. It is not suitable as a standalone paper; at most it merits an erratum/comment to the original arXiv note. The same graph already appears in the original paper as the first spider graph Ξ3\Xi_3, with recoverability of spider graphs proved there, so the novelty is essentially noticing that it contradicts Conjecture 2.

    Literature check: I found no later paper, correction, forum post, or note explicitly stating that Conjecture 2 is false or giving this counterexample. Searches around “antenna absorption,” “antenna jumping,” “punctured disk resistor networks,” and the paper title led back to the original arXiv paper only. The arXiv record appears to have only the 2018 version and no journal reference.

    Citation: Yulia Alexandr, Brian Burks, Sunita Chepuri, Patricia Commins, “Recovering Conductances of Resistor Networks in a Punctured Disk,” arXiv:1812.01517, especially Conjecture 2, Definition 25, Theorem 6, and Example 8.

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