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 $SAT(n,H)$ the set of all $H$-saturated graphs $G$ with order $n$. Then the saturation number $sat(n,H)$ is defined as ${min}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$, and the extremal number $ex(n,H)$ is defined as ${max}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$. A natural question is that of whether we can find an $H$-saturated graph with $m$ edges for any $sat(n,H)\le m\le ex(n,H)$. 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 $2\le i\le 6$. It is trivial for the case of $P_2$ that the saturated graph must be an empty graph.