1{2 "claims": [3 {4 "claim": "Zero forcing was introduced as a combinatorial upper bound for the maximum nullity M(G) of symmetric matrices described by a graph; M(G) <= Z(G) for every graph, and Z(Petersen graph) = 5.",5 "location": "intro",6 "sourceKeys": ["aim2008zeroforcing", "barioli2010parameters", "hogben2022book"],7 "kind": "literature"8 },9 {10 "claim": "The generalized Petersen graph P(n,k), n > 2k, has vertices u_i, v_i (i mod n) and edges u_i u_{i+1}, u_i v_i, v_i v_{i+k}; it is cubic on 2n vertices and its automorphism group was determined by Frucht, Graver and Watkins.",11 "location": "intro",12 "sourceKeys": ["watkins1969tait", "frucht1971groups"],13 "kind": "literature"14 },15 {16 "claim": "Z(P(n,k)) <= 2k+2 for all admissible n and k (Alameda et al. 2018, Theorem 2.2; reproved by Rashidi et al. 2020, Theorem 2.2); for k = 5 this gives Z(P(n,5)) <= 12.",17 "location": "lit",18 "sourceKeys": ["alameda2018families", "rashidi2020petersen", "arxiv2610_24059"],19 "kind": "literature"20 },21 {22 "claim": "Rashidi et al. 2020 give the refined upper bound Z(P(n,k)) <= r(s+2) when n = rk+s, 1 <= s <= k-1, the lower bound min{k, floor(n/k)} + 1 <= Z(P(n,k)) for k >= 3 via a K_{k,floor(n/k)} minor and the Colin de Verdiere parameter, and Z(P(2k+1,k)) = 6 for k >= 5, so Z(P(11,5)) = 6.",23 "location": "lit",24 "sourceKeys": ["rashidi2020petersen"],25 "kind": "literature"26 },27 {28 "claim": "Z(P(24r,5)) = 12 for all r >= 1 is reported by Rashidi et al. (Remark 2.4) as a theorem of Alameda et al. 2018, obtained there together with M(G) = Z(G) for certain generalized Petersen graphs; the primary statement was not accessible in this run.",29 "location": "lit",30 "sourceKeys": ["rashidi2020petersen", "alameda2018families"],31 "kind": "literature"32 },33 {34 "claim": "The published claim Z(P(n,3)) = 8 for n >= 12 is false at n = 12, where Z(P(12,3)) = 7; exhaustive search gives Z(P(n,3)) for 7 <= n <= 20 and confirms Z(P(11,5)) = 6.",35 "location": "lit",36 "sourceKeys": ["arxiv2607_19412", "rashidi2020petersen"],37 "kind": "literature"38 },39 {40 "claim": "Krishnan, Ma and Zhang prove Z(P(n,3)) = 8 for n >= 13 and Z(P(n,4)) = 10 for n >= 18 with sharp thresholds, using the external boundary of the first forcing sources, a column-deletion lemma valid for every k, forts, a Lean 4 formalization (k = 3) and DRAT-checked SAT certificates (k = 4); they state nothing about k = 5.",41 "location": "lit",42 "sourceKeys": ["arxiv2610_24059"],43 "kind": "literature"44 },45 {46 "claim": "A set S is a zero forcing set if and only if it intersects every fort (a nonempty set F such that no vertex outside F has exactly one neighbour in F); forts underlie integer-programming and fractional lower bounds for Z.",47 "location": "method",48 "sourceKeys": ["brimkov2019computational", "fast2018degrees", "cameron2023forts"],49 "kind": "literature"50 },51 {52 "claim": "Exact zero forcing numbers can be computed by integer programming or by a two-stage Boolean satisfiability formulation; SAT cardinality constraints use the sequential-counter encoding, solvers are driven through PySAT, and unsatisfiability proofs are checked with DRAT-trim.",53 "location": "method",54 "sourceKeys": ["brimkov2019computational", "brimkov2021improved", "sinz2005cardinality", "ignatiev2018pysat", "wetzler2014drattrim"],55 "kind": "literature"56 },57 {58 "claim": "The number of rounds of any forcing process is bounded by the propagation time, which gives a finite horizon for round-based encodings.",59 "location": "method",60 "sourceKeys": ["hogben2012propagation"],61 "kind": "literature"62 },63 {64 "claim": "For graphs of girth at least 5, Z(G) >= 2*delta - 2, which for cubic P(n,5) gives only Z >= 4.",65 "location": "lit",66 "sourceKeys": ["davila2015girth"],67 "kind": "literature"68 },69 {70 "claim": "Leaky forcing numbers of P(n,2) and P(n,3) for small n were computed by Bjorkman et al., who also exhibit the 2k+2-vertex zero forcing set {v_1,...,v_{2k}, u_k, u_{k+1}}; no k = 5 values appear there.",71 "location": "lit",72 "sourceKeys": ["bjorkman2025leaky"],73 "kind": "literature"74 },75 {76 "claim": "Lean 4 is the proof assistant used for any formal verification attempted in this study.",77 "location": "method",78 "sourceKeys": ["demoura2021lean4"],79 "kind": "literature"80 }81 ]82}83