KöMaL 2024 A888

Let $n$ be a given positive integer. Find the smallest positive integer $k$ for which the following statement is true: for any given simple connected graph $G$ and minimal cuts $V_1$, $V_2$, $\ldots$, $V_n$, at most $k$ vertices can be chosen with the property that picking any two of the chosen vertices there exists an integer $1\leq i\leq n$ such that $V_i$ separates the two vertices. A partition of the vertices of $G$ into two disjoint non-empty sets is called a minimal cut if the number of edges crossing the partition is minimal.