Some results about ID-path-factor critical graphs

Let G be a graph of order n . A spanning subgraph F of G is said to be a P ≥ k -factor of G if every component of F is a path with at least k vertices, where k ≥ 2 . In this paper, we introduce the concept of an ID-P ≥ k -factor critical graph; a graph G is said to be an ID-P ≥ k -factor critical graph if for any independent set I of G , G − I admits a P ≥ k -factor. We prove that a graph G of a given order is an ID-P ≥ 2 -factor critical graph if its binding number is at least 2 . We also prove that a graph G of a fixed order is an ID-P ≥ 3 -factor critical graph if its binding number is at least 94 . Furthermore, we show that the obtained results are the best possible in some sense.