  Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent
boundary
$\Gamma=\Gamma_0\cup\Gamma_1\cup\dots\cup\Gamma_m=:\Gamma_0\cup\tilde\Gamma$.
Let $u=u^f$ obey $\Delta u=0$ in $M$, $u|_{\Gamma_0}=f,\,\,u|_{\tilde\Gamma}=0$
(the grounded holes) and $v=v^h$ obey $\Delta v=0$ in $M$,
$v|_{\Gamma_0}=h,\,\,\partial_\nu v|_{\tilde\Gamma}=0$ (the isolated holes).
Let $\Lambda_{g}^{\rm gr}: f\mapsto\partial_\nu u^f|_{\Gamma_{0}}$ and
$\Lambda_{g}^{\rm is}: h\mapsto\partial_\nu v^h|_{\Gamma_{0}}$ be the
corresponding DN-maps. The EIT problem is to determine $M$ from
$\Lambda_{g}^{\rm gr}$ or $\Lambda_{g}^{\rm is}$.
  To solve it, an algebraic version of the BC-method is applied. The main
instrument is the algebra of holomorphic functions on the ma\-ni\-fold
${\mathbb M}$, which is obtained by gluing two examples of $M$ along
$\tilde{\Gamma}$. We show that this algebra is determined by $\Lambda_{g}^{\rm
gr}$ (or $\Lambda_{g}^{\rm is}$) up to isometric isomorphism. Its Gelfand
spectrum (the set of characters) plays the role of the material for
constructing a relevant copy $(M',g',\Gamma_{0}')$ of $(M,g,\Gamma_{0})$. This
copy is conformally equivalent to the original, provides
$\Gamma_{0}'=\Gamma_{0},\,\,\Lambda_{g'}^{\rm gr}=\Lambda_{g}^{\rm
gr},\,\,\Lambda_{g'}^{\rm is}=\Lambda_{g}^{\rm is}$, and thus solves the
problem.
