Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

The manuscript proves an exact pendant-forest reduction, a parity property for rooted-tree responses, a singular attachment lemma in the graph-port setting, and a rank-one criterion determining when a pendant leaf increases the line-graph signature. It establishes the universal upper bound s(L(G)) ≤ c(G). Together with the constructive lower bound, this gives floor((c+1)/2) ≤ f(c) ≤ c, so f(c) is finite and its maximum is attained for every cyclomatic number c. The sharper inequality 2s(L(G)) ≤ c(G)+1 remains a conjecture. The load-bearing mathematical certificates and principal computational results were independently checked. The repaired R2 graph6 package corrects labeled graph6 serialization; the previous edge-list calculations remain valid. Andrea Paone and Marco Paone contributed equally to this work. Status: Preprint; not peer reviewed.