A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two

A P≥3-factor F of a graph G is a spanning subgraph of G such that every component of F is a path of length at least two. Let R be a factor-critical graph with at least three vertices, that is, for each x ∈ V(R), R - x has a 1-factor (i.e., a perfect matching). Set V(R) = {x1, ...,xn}. Add new vertices {v1, ..., vn} to R together with the edges xivi, 1 ≤ i ≤ n. The resulting graph H is called a sun. (Note that degH vi = 1 for all i, 1≤i≤n.) K1 and K2, i.e., the complete graphs with one and two vertices, respectively, are also called suns. Then let C be the set of all suns. A sun component of a graph is a component which belongs to C. Let cs(G) denote the number of sun components of G. We prove that a graph G has a P≥3-factor if and only if cs(G - S)≤2|S|, for every subset S of V(G).