To prove that the connectivity $\kappa(H) = k$ for the graph $H$ obtained from a $k$-connected graph $G$ by adding a new vertex $w$ and connecting it to $k$ distinct vertices $v_1, v_2, \ldots, v_k$, we will use Menger's theorem and Whitney's inequalities.

Definitions and Setup

• A graph $G$ is $k$-connected if it has at least $k+1$ vertices, and the removal of fewer than $k$ vertices cannot disconnect the graph.
• Menger's theorem states that for any two non-adjacent vertices in a graph, the minimum number of vertices whose removal disconnects these vertices is equal to the maximum number of independent paths between them.
• Whitney's inequality for vertex connectivity states that for any graph $G$, $\kappa(G) \leq \delta(G)$, where $\kappa(G)$ is the vertex connectivity and $\delta(G)$ is the minimum degree of $G$.

Proof

1. Connectivity of $G$: Since $G$ is $k$-connected, it means that for any two vertices in $G$, there are at least $k$ vertex-disjoint paths between them. Additionally, the removal of fewer than $k$ vertices does not disconnect $G$.

2. Construction of $H$: The graph $H$ is obtained by adding a new vertex $w$ and $k$ edges $(w, v_i)$ for $i = 1, 2, \ldots, k$.

3. Minimum Degree in $H$: The minimum degree $\delta(H)$ of $H$ is at least $k$, because:

   • The new vertex $w$ has degree $k$.
   • Each of the vertices $v_1, v_2, \ldots, v_k$ has its degree increased by 1, making sure their degree is still at least $k$.

4. Applying Whitney's Inequality: By Whitney's inequality, $\kappa(H) \leq \delta(H)$. Since $\delta(H) \geq k$, we have: $\kappa(H) \leq k.$

5. Using Menger's Theorem: To show that $\kappa(H) \geq k$, we need to demonstrate that for any two vertices in $H$, there are at least $k$ vertex-disjoint paths between them.

   • Consider any two vertices in $H$ that are not $w$. Since $G$ is $k$-connected, there are at least $k$ vertex-disjoint paths between these two vertices in $G$, and these paths are preserved in $H$.
   • Consider $w$ and any vertex $v_i$. There are $k-1$ vertices in $\{v_1, v_2, \ldots, v_k\} \setminus \{v_i\}$ that provide $k-1$ disjoint paths, and the direct edge $(w, v_i)$ provides the $k$-th path. Therefore, there are $k$ vertex-disjoint paths between $w$ and $v_i$.

6. Combining Results: Since the conditions of Menger's theorem are satisfied, we conclude that $\kappa(H) \geq k$.

7. Conclusion: Combining the inequalities $\kappa(H) \leq k$ and $\kappa(H) \geq k$, we have: $\kappa(H) = k.$

Therefore, we have shown that the connectivity $\kappa(H)$ of the graph $H$ is indeed $k$.

Do you want details or have any questions?

Here are 5 numbered questions you could ask:

1. Can you provide an example of constructing $H$ from $G$ for a specific $k$?
2. How does Menger's theorem help in proving connectivity results?
3. What are some applications of Whitney's inequalities in graph theory?
4. How would the proof change if we were adding more than $k$ edges from $w$ to $G$?
5. What are the implications of this result in network design?

Tip: When dealing with graph connectivity problems, leveraging theorems like Menger's and inequalities like Whitney's can simplify the proofs significantly.