  The gravitational collapse of a barotropic perfect fluid having the Equation
of State (EoS) $p=k\rho$, where $k$ is constant, is studied here in the
framework of general relativity. We examine the restrictions on the
Misner-Sharp mass function, because of the introduction of such an EoS, in
terms of the compatibility of a certain pair of quasi-linear partial
differential equations, obtained from Einstein's field equations. We find that
except when this system of PDEs reduces to ODEs because of additional
symmetries imposed on the spacetimes, or when they become compatible with each
other in some special situations, consistent solution to perfect fluid collapse
with linear EoS is not available. The end state of collapse with no such
constraint of EoS has also been investigated. Since considering arbitrary
pressures in a collapsing cloud to study its end state is difficult as this
requires information about the dynamics of collapse not known in general, we
consider small perturbations to mass profiles corresponding to inhomogeneous
dust collapse. This, in turn, provides small pressure perturbations to the
otherwise pressureless fluid. The dependence of visibility or otherwise of the
singularity on initial conditions of collapse, in the presence or absence of
such perturbations is studied numerically. As long as no linear EoS is imposed
on the matter field, no incompatibility issue between its corresponding pair of
PDEs arise, unlike the case when $k$ is restricted to be a constant.
