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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 $n\\ge 7$ and size at most $\\lfloor 3n/2\\rfloor$ has an independent minimum vertex cut; (2) every connected graph of order $n$ with $n\\ge 7$ and size at most $2n$ has a foresty minimum vertex cut. 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