  Recently, it has been supposed that the Einstein-Gauss- Bonnet theory coupled
with scalar field (EGBS) maybe appropriately admit physically viable models of
celestial phenomena such that the scalar field effect is active in standard
four dimensions. We consider the spherically symmetric and static configuration
of the compact star and explain the consequences of the EGBS theory in the
frame of stellar modeling. In our formulation, for any given static profile of
the energy density $\rho$ with the spherical symmetry and the arbitrary
equation of state (EoS) of matter, we can construct the model which reproduces
the profile. Because the profile of the energy density determines the mass $M$
and the radius $R_s$ of the compact star, an arbitrary relation between the
mass $M$ and the radius $R_s$ of the compact star can be realized by adjusting
the potential and the coefficient function of the Gauss-Bonnet term in the
action of EGBS theory. This could be regarded as a degeneracy between the EoS
and the functions characterizing the model, which tells that only the
mass-radius relation is insufficient to constrain the model. For example, we
investigate a novel class of analytic spherically symmetric interior solutions
by the polytropic EoS. We discuss our model in detail and show that it is in
agreement with the necessary physical conditions required for any realistic
compact star approving that EGBS theory is consistent with observations.
