  We theoretically analyze the possibility to confine electrons in single-layer
graphene with the help of metallic gates, via the evaluation of the density of
states of such a gate-defined quantum dot in the presence of a ring-shaped
metallic contact. The possibility to electrostatically confine electrons in a
gate-defined ``quantum dot'' with finite-carrier density, surrounded by an
undoped graphene sheet, strongly depends on the integrability of the electron
dynamics in the quantum dot. With the present calculations we can
quantitatively compare confinement in dots with integrable and chaotic
dynamics, and verify the prediction that the Berry phase associated with the
pseudospin leads to partial confinement in situations where no confinement is
expected according to the arguments relying on the classical dynamics only.
