A graph G has a
$${\mathcal {P}}_{\geqslant k}$$
-factor if G has a spanning subgraph H such that every component of H is a path of order at least k. A graph G is
$${\mathcal {P}}_{\geqslant k}$$
-factor deleted if
$$G-e$$
has a
$${\mathcal {P}}_{\geqslant k}$$
-factor for each edge e of G. In this paper, we give two necessary and sufficient conditions defining a
$${\mathcal {P}}_{\geqslant 2}$$
-factor deleted graph and a
$${\mathcal {P}}_{\geqslant 3}$$
-factor deleted graph, respectively. Based on the result of
$${\ma...
