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

Abstract

A pseudo 2-factor of a graph is a spanning subgraph such that each component is $K_1$, $K_2$, 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 $\alpha(G)-\delta(G)+1$ components that are not cycles. For a graph $G$ and a set of vertices $S$, let $\delta_G(S)$ 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 $|I|-\delta_G(I)+1$ 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.