A note on the number of non-cycle components in a pseudo 2-factor of graphs

A pseudo 2-factor of a graph is a spanning subgraph such that each component is K₁, K₂, or a cycle. This notion was introduced by Bekkai and Kouider in 2009, where they showed that every graph G has a pseudo 2-factor with at most $α(G)-δ(G)+1$ components that are not cycles. For a graph G and a set of vertices S, let δG(S) denote the minimum degree of vertices in S. In this note, we show that every graph G has a pseudo 2-factor with at most $f(G)$ components that are not cycles, where $f(G)$ is the maximum value of |I|-δG(I)+1 among all independent sets I of G. This result is a common generalization of a result by Bekkai and Kouider and a previous result by the author on the existence of a 2-factor.