Sterboul’s theorem characterizes non-Kőnig–Egerváry graphs by the presence, relative to a maximum matching, of a flower or a posy. In this paper we translate that obstruction into the language of perfect flowers and the core of the graph. We introduce core-defective perfect flowers: perfect flowers whose alternating path contains a vertex at odd distance from the blossom base that does not belong to the core. We prove first that every Kőnig–Egerváry graph is core-rigid: in every perfect flower, all odd-distance vertices of the attaching path lie in the core. Conversely, if \(G\) is connected and is not an odd cycle, then \(G\) is non-Kőnig–Egerváry if and only if \(G\) contains a core-defective perfect flower. Thus, among connected graphs different from an odd cycle, the Kőnig–Egerváry graphs are exactly the graphs with no core-defective perfect flower. In the matchable case the statement strengthens: if \(G\) has a perfect matching, then being non-Kőnig–Egerváry is equivalent to the existence of a core-defective perfect flower for some maximum matching, and also equivalent to the existence of one for every maximum matching. We include examples and counterexamples showing why odd cycles, disconnected graphs, and the universal quantifier over maximum matchings require separate treatment.