We introduce a planted vertex cover problem on regular random graphs and study it by the cavity method. The equilibrium ordering phase transition of this binary-spin two-body interaction system is discontinuous in nature distinct from the continuous one of conventional Ising-like models, and it is dynamically blocked by an extensive free energy barrier. We discover that the disordered symmetric phase of this system may be locally stable with respect to the ordered phase at all inverse temperatures except for a unique eureka point $\beta_b$ at which it is only marginally stable. The eureka point $\beta_b$ serves as a backdoor to access the hidden ground state with vanishing free energy barrier. It exists in an infinite series of planted random graph ensembles and we determine their structural parameters analytically. The revealed new type of free energy landscape may also exist in other planted random-graph optimization problems at the interface of statistical physics and statistical inference.

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