Vertex connectivity of join of two graphs

Question

Does there exist any results on the vertex connectivity of join of two graphs?

If $G_1$ and $G_2$ are two graphs what can we say about the vertex connectivity of $G_1\lor G_2$ where $G_1\lor G_2$ is the join of $G_1$ and $G_2$? The join of $G_1$ and $G_1$ is the graph obtained from the disjoint union of $G_1$ and $G_2$ by adding all edges between $V(G_1)$ and $V(G_2)$.

If $\kappa(G)$ denotes vertex connectivity of a graph $G$ how is $\kappa(G_1\lor G_2)$ related to $\kappa(G_1)$ and $\kappa(G_2)$?

Answer 1

The vertex connectivity of $G_1 \vee G_2$ is $\min \{|V(G_1)|+\kappa(G_2), |V(G_2)|+\kappa(G_1)\}$. To see this, note that if $X \subseteq V(G_1 \vee G_2)$ is such that $V(G_1) \setminus X$ and $V(G_2) \setminus X$ are both non-empty, then $(G_1 \vee G_2) - X$ is connected. Thus, a smallest vertex separator $X$ of $G_1 \vee G_2$ must be of the form $V(G_1) \cup X_2$ or $V(G_2) \cup X_1$, where $X_i$ is a smallest vertex separator of $G_i$.