Let š be a set of connected graphs. Then a spanning subgraph A of G is called an š-factor if each component of A is isomorphic to some member of š. Especially, when every graph in š is a path, A is a path factor. For a positive integer d ā„ 2, we write š«ā„d = {š«i|i ā„ d}. Then a š«ā„d-factor means a path factor in which every component admits at least d vertices. A graph G is called a (š«ā„d,m)-factor deleted graph if G ā Eā² admits a š«ā„d-factor for any Eā² ā E(G) with |Eā²| = m. A graph G is called a (š«ā„d, k)-factor critical graph if G ā Q has a š«ā„d-factor for any Q ā V (G) with |Q| = k. In this paper, we present two degree conditions for graphs to be (š«ā„3,m)-factor deleted graphs and (š«ā„3, k)-factor critical graphs. Furthermore, we show that the two results are best possible in some sense.