The $\underline{\text{path number}}$ of a graph is the minimum number of paths we need to partition the vertices of a graph. Given a connected graph with the independence number $k > 1$, what is the maximum possible value for the path number in this graph? Find the answer in terms of $k$.

The independence number of a graph $\textbf{G}$ is the maximum possible number $k$, such that there exist $k$ pairwise non-adjacent vertices in $\textbf{G}$.
1. **Define the path number and independence number:**
   - The path number of a graph is the minimum number of paths needed to partition the vertices of the graph.
   - The independence number \( k \) of a graph \( G \) is the maximum number of pairwise non-adjacent vertices in \( G \).

2. **Claim:**
   - We claim that the maximum possible value for the path number in a connected graph with independence number \( k > 1 \) is \( k-1 \).

3. **Construct paths:**
   - Consider the largest path \( P \) in the graph \( G \) with vertices \( v_1, v_2, \ldots, v_l \).
   - Remove the vertices of \( P \) from \( G \) to get a subgraph \( G - \{v_1, v_2, \ldots, v_l\} \).
   - In this subgraph, find the largest path \( P_1 \) and continue this process to construct paths \( P_2, \ldots, P_s \) until all vertices are covered.

4. **Show \( s \le k-2 \):**
   - Let \( w_1, w_2, \ldots, w_s \) be the first vertices of paths \( P_1, \ldots, P_s \).
   - We need to show that \( v_1, v_l, w_1, \ldots, w_s \) form an independent set.
   - Since \( P \) is the largest path, \( v_1 \) and \( v_l \) do not have any adjacent vertices outside of \( P \).
   - For any \( 1 \le i < j \le s \), \( w_i \) and \( w_j \) cannot be adjacent because \( P_i \) is the largest path in the subgraph without vertices of \( P_1, P_2, \ldots, P_{i-1} \).

5. **Prove \( v_1 \) and \( v_l \) are not adjacent:**
   - Since \( G \) is connected, there exists a vertex \( w \) outside of \( P \) and a vertex \( v_i \) (\( 2 \le i \le l-1 \)) such that \( v_i \) and \( w \) are adjacent.
   - If \( v_1 \) and \( v_l \) were adjacent, the graph would have a path of length \( l+1 \), which contradicts the assumption that \( P \) is the largest path:
     \[
     w, v_i, v_{i-1}, \ldots, v_1, v_l, v_{l-1}, \ldots, v_{i+1}
     \]

6. **Conclusion:**
   - Since \( v_1, v_l, w_1, \ldots, w_s \) form an independent set with \( s+2 \) vertices, and the independence number is \( k \), we have \( s+2 \le k \).
   - Therefore, \( s \le k-2 \), and the path number of the graph is at most \( k-1 \).

7. **Construction of a graph with path number \( k-1 \):**
   - Consider a star graph with a central vertex \( v \) adjacent to \( k \) other vertices \( v_1, \ldots, v_k \).
   - This graph has an independence number \( k \) and a path number \( k-1 \).

\(\blacksquare\)

The final answer is \( \boxed{ k-1 } \)