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arXiv · 2607.19412

A correction to the Zero Forcing Number of the Generalized Petersen Graphs $P(n,3)$

Arnav Krishnan
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Abstract

Rashidi, Shajareh Poursalavati, and Tavakkoli [J. Algebra Comb. Discrete Struct. Appl. 7 (2020), no. 2, 183-193, Theorem 3.6] claim $Z(P(n,3)) = 8$ for all $n \geq 12$, but this fails at $n = 12$, where $Z(P(12,3)) = 7$. We provide an explicit 7-vertex zero forcing set for $P(12,3)$ with a fully traced forcing cascade, and confirm by exhaustive search that no 6-vertex set forces $P(12,3)$. We prove $Z(P(n,3)) \le 8$ for $n \ge 9$ using one explicit witness and symmetry. Exhaustive search yields $Z(P(n,3))$ for $7 \le n \le 20$ and identifies the gap in the published proof: its case analysis fails to exclude 7-vertex sets. We conjecture $Z(P(n,3)) = 8$ for $n \ge 13$; the missing ingredient is a lower-bound proof valid for all large $n$.

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Arnav Krishnan. 2026-07-13. A correction to the Zero Forcing Number of the Generalized Petersen Graphs $P(n,3)$. https://arxiv.org/abs/2607.19412

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