Sparse graphs with an independent or foresty minimum vertex cut

A connected graph is called fragile if it contains an independent vertex cut. In 2002 Chen and Yu proved that every connected graph of order n and size at most 2n−4 is fragile, and in 2013 Le and Pfender characterized the non-fragile graphs of order n and size 2n−3. It is natural to consider minimum vertex cuts. We prove two results. (1) Every connected graph of order n with n≥7 and size at most ⌊3n/2⌋ has an independent minimum vertex cut; (2) every connected graph of order n with n≥7 and size at most 2n has a foresty minimum vertex cut. Both results are best possible.