Geodesic
Connected (s,t)-Vertex Separator Parameterized by Chordality
Journal of Graph Algorithms and Applications, Tome 19 (2015) no. 1, pp. 549-565.

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We investigate the complexity of finding a minimum connected (s,t)-vertex separator ((s,t)-CVS) and present an interesting chordality dichotomy: we show that (s,t)-CVS is NP-complete on graphs of chordality at least 5 and present a polynomial-time algorithm for (s,t)-CVS on chordality 4 graphs. Further, we show that (s,t)-CVS is unlikely to have δlog2−εn-approximation algorithm, for any ε > 0 and for some δ > 0, unless NP has quasi-polynomial Las Vegas algorithms. On the positive-side of approximation, we present a ⎡c/2⎤-approximation algorithm for (s,t)-CVS on graphs with chordality c ≥ 3. Finally, in the parameterized setting, we show that (s,t)-CVS parameterized above the (s,t)-vertex connectivity is W[2]-hard. 
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     author = {N. Narayanaswamy and N. Sadagopan},
     title = {Connected {(s,t)-Vertex} {Separator} {Parameterized} by {Chordality}},
     journal = {Journal of Graph Algorithms and Applications},
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N. Narayanaswamy; N. Sadagopan. Connected (s,t)-Vertex Separator Parameterized by Chordality. Journal of Graph Algorithms and Applications, Tome 19 (2015) no. 1, pp. 549-565. doi : 10.7155/jgaa.00377. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00377/

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