  A spanning subgraph $F$ of $G$ is called a path factor if every component of
$F$ is a path of order at least 2. Let $k\geq2$ be an integer. A $P_{\geq
k}$-factor of $G$ means a path factor in which every component has at least $k$
vertices. A graph $G$ is called a $P_{\geq k}$-factor avoidable graph if for
any $e\in E(G)$, $G$ has a $P_{\geq k}$-factor avoiding $e$. A graph $G$ is
called a $(P_{\geq k},n)$-factor critical avoidable graph if for any
$W\subseteq V(G)$ with $|W|=n$, $G-W$ is a $P_{\geq k}$-factor avoidable graph.
In other words, $G$ is $(P_{\geq k},n)$-factor critical avoidable if for any
$W\subseteq V(G)$ with $|W|=n$ and any $e\in E(G-W)$, $G-W-e$ admits a $P_{\geq
k}$-factor. In this article, we verify that (\romannumeral1) an
$(n+r+2)$-connected graph $G$ is $(P_{\geq2},n)$-factor critical avoidable if
$I(G)>\frac{n+r+3}{2(r+2)}$; (\romannumeral2) an $(n+r+2)$-connected graph $G$
is $(P_{\geq3},n)$-factor critical avoidable if $t(G)>\frac{n+r+2}{2(r+2)}$;
(\romannumeral3) an $(n+r+2)$-connected graph $G$ is $(P_{\geq3},n)$-factor
critical avoidable if $I(G)>\frac{n+3(r+2)}{2(r+2)}$; where $n$ and $r$ are two
nonnegative integers.
