  This paper is concerned with the variational problem for the elastic energy
defined on symmetric graphs under the unilateral constraint. Assuming that the
obstacle function satisfies the symmetric cone condition, we prove (i)
uniqueness of minimizers, (ii) loss of regularity of minimizers, and give (iii)
complete classification of existence and non-existence of minimizers in terms
of the size of obstacle. As an application, we characterize the solution of the
obstacle problem as equilibrium of the corresponding dynamical problem.
