Summary: 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 \(k \geq 2\) 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 (i) an \((n + r + 2)\)-connected graph \(G\) is \((P_{\geq 2}, n)\)-factor critical avoidable if \(I(G)> \frac{n+r+3}{2(r+2)}\); (ii) an \((n + r + 2)\)-connected graph \(G\) is \((P_{\geq 3}, n)\)-factor critical avoidable if \(t(G)>\frac{n+r+2}{2(r+2)}\); (iii) an \((n + r + 2)\)-connected graph \(G\) is \((P_{\geq 3}, n)\)-factor critical avoidable if \(I(G)>\frac{n+3(r+2)}{2(r+2)}\); where \(n\) and \(r\) are two nonnegative integers.