On the saturation number of the kite graph

For a fixed graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain a copy of $H$, but adding any edge $e \in E(\overline{G})$ to $G$ creates a copy of $H$. The saturation number $\mathrm{sat}(n,H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. Let $K$ be the kite graph, formed by removing one edge from $ K_4$ and then attaching a pendant edge to a vertex of degree two in the resulting graph.In this paper, we first establish a relationship between connectivity and $K$-saturated graphs, and subsequently determine the saturation number of the kite graph $K$. Moreover, we completely characterize all extremal graphs.Our result provides a partial answer to a problem raised by Hua and Peng [Discrete Math. 349 (2026) 114674].