Saturation numbers for joins of graphs and characterization of extremal graphs

A graph $G$ is $H$-saturated if $G$ contains no $H$-copy as a subgraph, but adding any edge between two non-adjacent vertices in $G$ creates a copy of $H$. The saturation number $\mathrm{s}\mathrm{a}\mathrm{t}(n,H)$ is the minimum number of edges in an $n$-vertex $H$-saturated graph. Saturation number for the join of a vertex and a graph $F$, denoted by $K_1\vee F$, has received considerable attention. Cameron and Puleo [Discrete Math. 345 (2022), 112867] showed that $\mathrm{s}\mathrm{a}\mathrm{t}(n,K_1\vee F)\le n-1+\mathrm{s}\mathrm{a}\mathrm{t}(n-1,F)$ for all $n>\mathrm{\mid }V(F)\mathrm{\mid }$. A natural question is to ask when the above equality holds. Existing results for $\mathrm{s}\mathrm{a}\mathrm{t}(n,K_1\vee F)$ always constrain that a non-empty graph $F$ contains no isolated vertex. In this paper, we investigate the saturation number of $K_1\vee F$ when a non-empty graph $F$ contains an isolated vertex. We first determine the saturation number for $K_1\vee F$ when $F=K_{p-1}\cup K_1$. When $p=3$, we extend the result to any number of isolated vertices, and determine the saturation number for $K_1\vee F$ when $F=K_2\cup q{K}_1$, or $F=2K_2\cup q{K}_1$ for any $q\ge 1$. Moreover, all minimum saturated graphs are fully characterized. In our results, $\mathrm{s}\mathrm{a}\mathrm{t}(n,K_1\vee F)=n-1+\mathrm{s}\mathrm{a}\mathrm{t}(n-1,F)$ holds when $F=K_2\cup q{K}_1$, or $F=2K_2\cup q{K}_1$ for any $q\ge 1$; but fails when $F=K_{p-1}\cup K_1$ for $p\ge 4$.