  The generalized constrained search formalism is used to address the issues
concerning density-to-potential mapping for excited states in time-independent
density-functional theory. The multiplicity of potentials for any given density
and the uniqueness in density-to-potential mapping are explained within the
framework of unified constrained search formalism for excited-states due to
G\"orling, Levy-Nagy, Samal-Harbola and Ayers-Levy. The extensions of
Samal-Harbola criteria and it's link to the generalized constrained search
formalism are revealed in the context of existence and unique construction of
the density-to-potential mapping. The close connections between the proposed
criteria and the generalized adiabatic connection are further elaborated so as
to keep the desired mapping intact at the strictly correlated regime.
Exemplification of the unified constrained search formalism is done through
model systems in order to demonstrate that the seemingly contradictory results
reported so far are neither the true confirmation of lack of Hohenberg-Kohn
theorem nor valid representation of violation of Gunnarsson-Lundqvist theorem
for excited states. Hence the misleading interpretation of subtle differences
between the ground and excited state density functional formalism are
exemplified.
