Saturation numbers for joins of graphs and characterization of extremal graphs

Abstract: 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 $\mathrm{sat}(n,H)$ 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 \cite{Ca} showed that $$\mathrm{sat}(n,K_1 \vee F)\le n-1+\mathrm{sat}(n-1, F)$$ for all $n > |V(F)|$. A natural question is to ask when the above equality holds. Existing results for $\mathrm{sat}(n,K_1 \vee F)$ 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 $F=K_{2}\cup qK_1$, or $F=2K_{2}\cup qK_1$ for any $q\ge 1$. Moreover, all minimum saturated graphs are fully characterized. In our results, $$\mathrm{sat}(n,K_1 \vee F)= n-1+\mathrm{sat}(n-1, F)$$ holds when $F=K_2\cup qK_1$, or $F=2K_2\cup qK_1$ for any $q\ge 1$; but fails when $F=K_{p-1}\cup K_1$ for $p\ge 4$.