On the Complexity of Connected (s, t)-Vertex Separator

We show that minimum connected (s, t)-vertex separator ((s, t)-CVS) is Ω(log2− n)-hard for any > 0 unless NP has quasi-polynomial Las-Vegas algorithms. i.e., for any > 0 and for some δ > 0, (s, t)-CVS is unlikely to have δ.log2− n-approximation algorithm. We show that (s, t)-CVS is NPcomplete on graphs with chordality at least 5 and present a polynomial-time algorithm for (s, t)-CVS on bipartite chordality 4 graphs. We also present a d c 2 e-approximation algorithm for (s, t)-CVS on graphs with chordality c. Finally, from the parameterized setting, we show that (s, t)-CVS parameterized above the (s, t)-vertex connectivity is W [2]-hard.