The edge spectrum of the saturation number for small paths

Let H be a simple graph. A graph G is called an H -saturated graph if H is not a subgraph of G , but adding any missing edge to G will produce a copy of H . Denote by S A T ( n , H ) the set of all H -saturated graphs G with order n . Then the saturation number s a t ( n , H ) is defined as min G ∈ S A T ( n , H ) | E ( G ) | , and the extremal number e x ( n , H ) is defined as max G ∈ S A T ( n , H ) | E ( G ) | . A natural question is that of whether we can find an H -saturated graph with m edges for any s a t ( n , H ) ≤ m ≤ e x ( n , H ) . The set of all possible values m is called the edge spectrum for H -saturated graphs. In this paper we investigate the edge spectrum for P i -saturated graphs, where 2 ≤ i ≤ 6 . It is trivial for the case of P 2 that the saturated graph must be an empty graph.