We show that minimum connected $(s,t)$-vertex separator ($(s,t)$-CVS) is
$\Omega(log^{2-\epsilon}n)$-hard for any $\epsilon >0$ unless NP has
quasi-polynomial Las-Vegas algorithms. i.e., for any $\epsilon >0$ and for some
$\delta >0$, $(s,t)$-CVS is unlikely to have
$\delta.log^{2-\epsilon}n$-approximation algorithm. We show that $(s,t)$-CVS is
NP-complete on graphs with chordality at least 5 and present a polynomial-time
algorithm for $(s,t)$-CVS on bipartite chordality 4 graphs.