风筝图的饱和数
On the saturation number of the kite graph
AI总结：
本文研究风筝图的饱和数，建立其与连通性的关系，确定该饱和数并刻画极值图，部分回答了Hua和Peng的相关问题。
AI中文摘要：
对于一个固定图H，若图G不含H的复制图，但向G的补图的任意边e添加到G中都会生成H的复制图，则称G是H-饱和图。饱和数sat(n,H)是n个顶点的H-饱和图中边的最小数量。令K为风筝图，其构造方式为：从完全图K₄中移除一条边，再向所得图中一个度数为2的顶点附加一条悬边。本文中，我们首先建立连通性与K-饱和图之间的关系，随后确定风筝图K的饱和数，还完整刻画了所有极值图。该结果部分回答了Hua和Peng提出的问题[《离散数学》349卷（2026），114674]。
英文摘要：
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].
