What does this research mean for the field?

Static counting information and bounded-cost reachability capture distinct features of a solution space, as evidenced by graphs with identical vertex-cover counting polynomials that exhibit different reconfiguration peak costs. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine whether boundary-conditioned vertex-cover counting polynomials capture the bounded-cost reachability and peak costs needed to reconfigure graph covers.

September 8, 2026Open Access

Boundary Counts Do Not Determine Cover Reconfiguration

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To determine whether boundary-conditioned vertex-cover counting polynomials capture the bounded-cost reachability and peak costs needed to reconfigure graph covers.
Constructed a four-cycle and a paw graph sharing identical boundary-conditioned vertex-cover counting polynomials under one-vertex gluing.
Evaluated token-addition and removal transitions between minimum vertex covers to determine exact intermediate peak costs.
Analyzed an integer hypergraph based on unit-fraction equations (1/x = 1/y + 1/z) and verified reconfiguration bounds using finite algorithmic certificates.
Showed that despite identical rooted cover-size distributions, the four-cycle and paw require token-addition peak costs of four and three, respectively.
Proved that the Diophantine hypergraph with costs a ≥ b > 0 achieves an optimum cost of b floor(2k/3) and an exact reconfiguration peak of b floor(2k/3) + b.
Identified a weighted instance outside the primary parameter regime that yields an optimum cost of six alongside a peak cost of ten.

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Implication

Mathematical analysis demonstrates divergence between boundary counting polynomials and reconfiguration peak costs in graph covers, indicating limits of static counting.

Cite This Study

Seth Christopher (2026) studied this question.

synapsesocial.com/papers/6a9fd7ab58e84d0ff5b46878 https://doi.org/10.5281/zenodo.22571896

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Implication

Mathematical analysis demonstrates divergence between boundary counting polynomials and reconfiguration peak costs in graph covers, indicating limits of static counting.

Boundary Counts Do Not Determine Cover Reconfiguration

Key Points

Discussion

Implication

Cite This Study

Boundary Counts Do Not Determine Cover Reconfiguration

Key Points

Discussion

Implication

Cite This Study

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