Computing the Odd-Ear Prefix Parameter Is NP-Complete

For a 2-vertex-connected graph G, let beta(G) be the maximum number of initial odd ears in an open ear decomposition, counting its initial cycle. We prove that deciding beta(G) >= k, with k part of the input, is NP-complete. This gives a complexity classification for a question of Benchetrit and Sebo. Hardness holds even for simple, 2-connected, 2-degenerate graphs whose deletion of one vertex is bipartite, and whose girth is at least any prescribed fixed bound. The proof identifies beta on a cone over a bipartite multigraph with the largest induced edge count on a connected vertex set admitting a perfect matching. Two polynomially bounded edge multiplicities adapt a known connected-matching satisfiability gadget to this different objective. Odd subdivision removes parallel edges without changing beta. The known gadget and theorem inputs are credited. This is an AI-assisted, self-audited, unrefereed preprint; no independent verification or absolute priority claim is made.