New Invariants for Partitioning a Graph into 2-connected Subgraphs

Abstract

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.