Some results on minimum saturated graphs

Let $G$ be a graph and $\mathcal{F}$ be a family of graphs. We say a graph $G$ is $\mathcal{F}$-saturated if $G$ does not contain any member in $\mathcal{F}$ and for any $e\in E(\bar{G})$, $G+e$ creates a copy of some member in $\mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges of an $\mathcal{F}$-saturated graphs with $n$ vertices, denoted by $\text{sat}(n,\mathcal{F})$. If $\mathcal{F}=\{F\}$, then we write it as $\text{sat}(n,F)$ for short. In this paper, we determine the exact value of $\text{sat}(n,\{K_3,P_k\})$, and as its application, we obtain two bounds of $\text{sat}(n,K_3\cup P_k)$ for $k\ge 10$ and sufficiently large $n$. Furthermore, $\text{sat}(n,K_1\vee F)$ is determined, where $F$ is a linear forest without isolated vertices.