  Given a set $P$ of $n$ points in the plane, the $k$-center problem is to find
$k$ congruent disks of minimum possible radius such that their union covers all
the points in $P$. The $2$-center problem is a special case of the $k$-center
problem that has been extensively studied in the recent past \cite{CAHN,HT,SH}.
In this paper, we consider a generalized version of the $2$-center problem
called \textit{proximity connected} $2$-center (PCTC) problem. In this problem,
we are also given a parameter $\delta\geq 0$ and we have the additional
constraint that the distance between the centers of the disks should be at most
$\delta$. Note that when $\delta=0$, the PCTC problem is reduced to the
$1$-center(minimum enclosing disk) problem and when $\delta$ tends to infinity,
it is reduced to the $2$-center problem. The PCTC problem first appeared in the
context of wireless networks in 1992 \cite{ACN0}, but obtaining a nontrivial
deterministic algorithm for the problem remained open. In this paper, we
resolve this open problem by providing a deterministic $O(n^2\log n)$ time
algorithm for the problem.
