Connectivity keeping paths in k-connected graphs

Abstract

A result of G. Chartrand, A. Kaugars, and D. R. Lick [Proc Amer Math Soc 32(1972), 63-68] says that every finite, k-connected graph G of minimum degree at least ⌊3k/2⌋ contains a vertex x such that G-x is still k-connected. We generalize this result by proving that every finite, k-connected graph G of minimum degree at least ⌊3k/2⌋ + m-1 for a positive integer m contains a path Pof length m-1 such that G-V(P) is still k-connected. This has been conjectured in a weaker form by S. Fujita and K. Kawarabayashi [J Combin Theory Ser B 98(2008), 805-811]. © 2009 Wiley Periodicals, Inc.