  Let $\mathcal{A}$ be a set of connected graphs. Then a spanning subgraph $A$
of $G$ is called an $\mathcal{A}$-factor if each component of $A$ is isomorphic
to some member of $\mathcal{A}$. Especially, when every graph in $\mathcal{A}$
is a path, $A$ is a path factor. For a positive integer $d\geq2$, we write
$\mathcal{P}_{\geq d}=\{P_i|i\geq d\}$. Then a $\mathcal{P}_{\geq d}$-factor
means a path factor in which every component admits at least $d$ vertices. A
graph $G$ is called a $(\mathcal{P}_{\geq d},m)$-factor deleted graph if $G-E'$
admits a $\mathcal{P}_{\geq d}$-factor for any $E'\subseteq E(G)$ with
$|E'|=m$. A graph $G$ is called a $(\mathcal{P}_{\geq d},k)$-factor critical
graph if $G-Q$ has a $\mathcal{P}_{\geq d}$-factor for any $Q\subseteq V(G)$
with $|Q|=k$. In this paper, we present two degree conditions for graphs to be
$(\mathcal{P}_{\geq3},m)$-factor deleted graphs and
$(\mathcal{P}_{\geq3},k)$-factor critical graphs. Furthermore, we show that the
two results are best possible in some sense.
