  In random geometry, a recurring theme is that any two geodesics emanating
from a typical point part ways at a strictly positive distance from the above
point, and we call such points as $1$-stars. However, the measure zero set of
atypical stars, the points where such coalescence fails, is typically
uncountable and the corresponding Hausdorff dimensions of these sets have been
heavily investigated for a variety of models including the directed landscape,
Liouville quantum gravity and the Brownian map. In this paper, we consider the
directed landscape -- the scaling limit of last passage percolation as
constructed in the work Dauvergne-Ortmann-Vir\'ag '18 -- and look into the
Hausdorff dimension of the set of atypical stars lying on a geodesic. We show
that the above dimension is almost surely equal to $1/3$. This is in contrast
to Ganguly-Zhang '22, where it was shown that set of atypical stars on the line
$\{x=0\}$ has dimension $2/3$. This reduction of the dimension from $2/3$ to
$1/3$ yields a quantitative manifestation of the smoothing of the environment
around a geodesic with regard to exceptional behaviour.
