1. The problem is to find the diameter of a generalized Petersen graph. 2. A generalized Petersen graph $GP(n,k)$ is defined for integers $n \geq 3$ and $1 \leq k < \frac{n}{2}$, with vertices and edges arranged in a specific pattern. 3. The diameter of a graph is the greatest distance between any pair of vertices, where distance is the length of the shortest path connecting them. 4. For the generalized Petersen graph $GP(n,k)$, the diameter depends on $n$ and $k$ and is known to satisfy certain bounds. 5. Specifically, the diameter $D$ of $GP(n,k)$ is at most $\left\lfloor \frac{n}{2} \right\rfloor + 1$. 6. In many cases, the diameter is exactly $\left\lfloor \frac{n}{2} \right\rfloor$ or $\left\lfloor \frac{n}{2} \right\rfloor + 1$, depending on the values of $n$ and $k$. 7. To find the exact diameter for given $n$ and $k$, one typically analyzes the structure or uses known results from graph theory literature. 8. For example, the Petersen graph itself is $GP(5,2)$ and has diameter 2. 9. Thus, the diameter of a generalized Petersen graph $GP(n,k)$ is generally about half of $n$, specifically bounded by $D \leq \left\lfloor \frac{n}{2} \right\rfloor + 1$. Final answer: The diameter of $GP(n,k)$ satisfies $$D \leq \left\lfloor \frac{n}{2} \right\rfloor + 1.$$