Saturation numbers of $K_2\vee P_k$

A graph $G$ is called $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e\in E(\bar{G})$. The saturation number of $H$ is the minimum number of edges in an $H$-saturated graph of order $n$, denoted by $sat(n,H)$. In this paper, we investigate $sat(n,K_2\vee P_k)$, where $k\ge 3$. Let $a_k$ be an integer, defined as follows: $a_k=k$ for $3\le k\le 5$; $a_k=3\cdot 2^{t-1}-2$ for $k=2t\ge 6$; and $a_k=2^{t+1}-2$ for $k=2t+1\ge 7$. We show that $sat(n,K_2\vee P_k)=2n-3+sat(n-2,P_k)$ for $n\ge a_k+2$ and $k\ge 3$, characterize the $K_2\vee P_k$-saturated graphs with $sat(n,K_2\vee P_k)$ edges, the $K_1\vee P_k$-saturated graphs with $sat(n,K_1\vee P_k)$ edges for $3\le k\le 5$ and the $P_k$-saturated graphs with $sat(n,P_k)$ edges for $3\le k\le 4$. Furthermore, we propose some questions for further research.