  A spanning subgraph $H$ of a graph $G$ is called a $P_{\geq k}$-factor of $G$
if every component of $H$ is isomorphic to a path of order at least $k$, where
$k\geq2$ is an integer. A graph $G$ is called a $(P_{\geq k},l)$-factor
critical graph if $G-V'$ contains a $P_{\geq k}$-factor for any $V'\subseteq
V(G)$ with $|V'|=l$. A graph $G$ is called a $(P_{\geq k},m)$-factor deleted
graph if $G-E'$ has a $P_{\geq k}$-factor for any $E'\subseteq E(G)$ with
$|E'|=m$. Intuitively, if a graph is dense enough, it will have a $P_{\geq
3}$-factor. In this paper, we give some sufficient conditions for a graph to be
a $(P_{\geq 3},l)$-factor critical graph or a $(P_{\geq 3},m)$-factor deleted
graph. In this paper, we demonstrate that (i) $G$ is a $(P_{\geq 3},l)$-factor
critical graph if its sun toughness $s(G)>\frac{l+1}{3}$ and $\kappa(G)\geq
l+2$. (ii) $G$ is a $(P_{\geq 3},l)$-factor critical graph if its degree sum
$\sigma_3(G)\geq n+2l$ and $\kappa(G)\geq l+1$. (iii) $G$ is a $(P_{\geq
3},m)$-factor deleted graph if its sun toughness $s(G)\geq \frac{m+1}{m+2}$ and
$\kappa(G)\geq 2m+1$. (iv) $G$ is a $(P_{\geq 3},m)$-factor deleted graph if
its degree sum $\sigma_3(G)\geq n+2m$ and $\kappa(G)\geq 2m+1$.
