  We investigate the breathing mode and the stability of a quantum droplet in a
tightly trapped one-dimensional dipolar gas of bosonic atoms. When the droplet
with a flat-top density profile is formed, the breathing mode frequency scales
as the inverse of the number of atoms in the cloud. This is straightforwardly
derived within a phenomenological hydrodynamical approach and confirmed using
both a variational method based on a generalized Gross-Pitaevskii action
functional and the sum-rule approach. We extend our analysis also to the
presence of axial confinement showing the effect of the trap on the density
profile and therefore on the breathing mode frequency scaling. Our analysis
confirms the stability of the quantum droplet against the particles emission
when the flat-top density profile is observed. Our results can be used as a
guide to the experimental investigations of collective modes to detect the
formation of quantum droplets in quasi-one-dimensional dipolar gases.
