Determine Vertex Connectivity of Graph

Given a graph G, determine its vertex connectivity $\kappa(G)$ by identifying the minimum number of vertices that can be deleted to disconnect the graph or make it trivial.

For example, in specific cases, identify if deleting certain vertices (e.g., $V_2$ or $V_5, V_7$) disconnects the graph or reduces it to a trivial graph.

Understanding the concept of vertex connectivity is crucial in the field of graph theory as it measures the robustness or the vulnerability of a graph to the removal of vertices. Vertex connectivity, represented by kappa of G, is the minimum number of vertices that need to be removed to either disconnect the graph or reduce it to a single vertex. This concept finds applications in network design, where ensuring connectivity despite failures or attacks is a fundamental requirement.

When tackling problems involving vertex connectivity, one strategy is to examine various subsets of vertices to determine if their removal disconnects the graph. This involves a blend of intuition—often informed by understanding the graph's structure—and systematic examination. For instance, recognizing articulation points, which are vertices that increase the number of connected components when removed, can provide valuable insights.

In practice, calculating vertex connectivity often involves algorithms or heuristic methods especially in complex or large-scale graphs. However, in simpler or well-structured graphs, it might be straightforward enough to visually inspect and identify critical vertices. This problem challenges your ability to think combinatorially and apply your knowledge of graph structures, and how different types of graphs, such as complete graphs or cycles, inherently possess certain connectivity attributes.