Degree conditions for path-factor critical deleted or covered graphs

A path-factor of a graph G is a spanning subgraph of G whose components are paths. A P≥d-factor of a graph G is a path-factor of G whose components are paths with at least d vertices, where d is an integer with d ≥ 2. A graph G is P≥d-factor covered if for any e ∈ E(G), G admits a P≥d-factor including e. A graph G is (P≥d,n)-factor critical deleted if for any Q ⊆ V (G) with |Q| = n and any e ∈ E(G − Q), G − Q − e has a P≥d-factor. A graph G is (P≥d,n)-factor critical covered if for any Q ⊆ V (G) with |Q| = n, G − Q is a P≥d-factor covered graph. In this paper, we verify that (i) an (n + t + 2)-connected graph G of order p with p ≥ 4t + n + 7 is (P≥3,n)-factor critical deleted if
 for any independent set {v1,v2,··· ,v2t+1} of G, where