Minimizing the number of edges in ${\mathcal{C}}_{\ge r}$-saturated graphs

Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in \mathcal{F}$ but the addition of any edge $e\mathrm{\notin }E(G)$ creates at least one copy of some $F\in \mathcal{F}$ within $G$. The minimum size of an $\mathcal{F}$-saturated graph on $n$ vertices are called the saturation number, denoted by $\text{sat}(n,\mathcal{F})$. Let ${\mathcal{C}}_{\ge r}$ be the family of cycles of length at least $r$. Ferrara et al. (2012) gave lower and upper bounds of $\text{sat}(n,C_{\ge r})$ and determined the exact values of $\sat(n, C_{\ge r})$for$3\le r\le 5$\mathrm{.}Inthispaper,wedeterminetheexactvalueof$\sat(n,$\mathcal{C}$_{\ge r})$for$r=6$and$28\le \frac{n}2\le r\le n$ and give new upper and lower bounds for the other cases.