  We report for the first time a parametric fit to the pattern of the \ell = 1
mixed modes in red giants, which is a powerful tool to identify
gravity-dominated mixed modes. With these modes, which share the
characteristics of pressure and gravity modes, we are able to probe directly
the helium core and the surrounding shell where hydrogen is burning. We propose
two ways for describing the so-called mode bumping that affects the frequencies
of the mixed modes. Firstly, a phenomenological approach is used to describe
the main features of the mode bumping. Alternatively, a quasi-asymptotic
mixed-mode relation provides a powerful link between seismic observations and
the stellar interior structure. We used period \'echelle diagrams to emphasize
the detection of the gravity-dominated mixed modes. The asymptotic relation for
mixed modes is confirmed. It allows us to measure the gravity-mode period
spacings in more than two hundred red giant stars. The identification of the
gravity-dominated mixed modes allows us to complete the identification of all
major peaks in a red giant oscillation spectrum, with significant consequences
for the true identification of \ell = 3 modes, of \ell = 2 mixed modes, for the
mode widths and amplitudes, and for the \ell = 1 rotational splittings. The
accurate measurement of the gravity-mode period spacing provides an effective
probe of the inner, g-mode cavity. The derived value of the coupling
coefficient between the cavities is different for red giant branch and clump
stars. This provides a probe of the hydrogen-shell burning region that
surrounds the helium core. Core contraction as red giants ascend the red giant
branch can be explored using the variation of the gravity-mode spacing as a
function of the mean large separation.
