  A vertex partition in which every part induces a 2-connected subgraph is
called a 2-proper partition. This concept was introduced by Ferrara et al. in
2013, and Borozan et al. gave the best possible minimum degree condition for
the existence of a 2-proper partition in 2016. Later, in 2022, Chen et al.
extended the result by showing a minimum degree sum condition for the existence
of 2-proper partition. In this paper, we introduce two new invariants of graph,
denoted by $\sigma^*(G)$ and $\alpha^*(G)$. These two invariants are defined
from degree sum on all independent sets with some property. We prove that if a
graph $G$ satisfies $\sigma^*(G)\geq |V(G)|$, then with some exceptions, $G$
has a 2-proper partition with at most $\alpha^*(G)$ parts. This result is best
possible, and implies both of the results by Borozan et al. and by Chen et al..
Moreover, as a corollary of our result, we give a minimum degree product
condition for the existence of a 2-proper partition.
