  The question of self-adjoint realizations of sign-indefinite second order
differential operators is discussed in terms of a model problem. Operators of
the type $-\frac{d}{dx} \sgn (x) \frac{d}{dx}$ are generalized to finite, not
necessarily compact, metric graphs. All self-adjoint realizations are
parametrized using methods from extension theory. The spectral and scattering
theory of the self-adjoint realizations are studied in detail.
