CGMO 2009 Problem 4

Let $n$ be an integer greater than $3.$ Points $V_{1},V_{2},...,V_{n},$ with no three collinear, lie on a plane. Some of the segments $V_{i}V_{j},$ with $1 \le i < j \le n,$ are constructed. Points $V_{i}$ and $V_{j}$ are neighbors if $V_{i}V_{j}$ is constructed. Initially, chess pieces $C_{1},C_{2},...,C_{n}$ are placed at points $V_{1},V_{2},...,V_{n}$ (not necessarily in that order) with exactly one piece at each point. In a move, one can choose some of the $n$ chess pieces, and simultaneously relocate each of the chosen piece from its current position to one of its neighboring positions such that after the move, exactly one chess piece is at each point and no two chess pieces have exchanged their positions. A set of constructed segments is called harmonic if for any initial positions of the chess pieces, each chess piece $C_{i}(1 \le i \le n)$ is at the point $V_{i}$ after a finite number of moves. Determine the minimum number of segments in a harmonic set.