Saturation numbers of $K_3 \vee P_k$

Source abstract

A graph $G$ is said to be $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e$ of $\overline{G}$. The saturation number of $H$, denoted by $\mathrm{sat}(n, H)$, is the minimum number of edges among all $H$-saturated graphs of order $n$. In this paper, we investigate the saturation number of $K_3 \vee P_k$ for $k \ge 3$ and show that $\mathrm{sat}(n, K_3 \vee P_k) = 3n - 6 + \mathrm{sat}(n-3, P_k)$ holds for all $n \ge a_k + 3$ and $k \ge 3$, which verifies the conjecture [Discrete Appl. Math, 394(2026) 62-80] on $\mathrm{sat}(n, K_s \vee P_k)$ is true for the case $s=3$, where $a_k = k$ 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 $k = 2t + 1 \ge 7$.