Geodesic
Removing vertices from $k$-connected graphs without losing $k$-connectivity
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part I, Tome 340 (2006), pp. 103-116

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The problem of removing vertices from a $k$-connected graph without losing $k$-connectivity is studied. We prove that one can remove some inner vertices from $k$-blocks, provided the interior of each block is large enough with respect to its boundary and the degree of any vertex of the graph is greater than $\frac{3k-1}{2}$ or $\frac{3k}{2}$. 
@article{ZNSL_2006_340_a6,
     author = {A. S. Chukhnov},
     title = {Removing vertices from $k$-connected graphs without losing $k$-connectivity},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {103--116},
     publisher = {mathdoc},
     volume = {340},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_340_a6/}
}
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A. S. Chukhnov. Removing vertices from $k$-connected graphs without losing $k$-connectivity. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part I, Tome 340 (2006), pp. 103-116. http://geodesic.mathdoc.fr/item/ZNSL_2006_340_a6/