  In this paper, we consider an inverse conductivity problem on a bounded
domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, also known as Electrical
Impedance Tomography (EIT), for the case where unknown impenetrable obstacles
are embedded into $\Omega$. We show that a piecewise-constant conductivity
function and embedded obstacles can be simultaneously recovered in terms of the
local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the
boundary of the domain $\Omega$. The method depends on the well-posedness of a
coupled PDE-system constructed for the conductivity equations in the
$H^1$-space and some elementary a priori estimates for Harmonic functions.
