17. In the plane, 7 points are given, connect them with some line segments so that
(1) among any three points, at least two are connected,
(2) the number of line segments is minimized.

How many line segments are there? Provide such a graph.
17. The figure below shows that 9 line segments are sufficient.

Now prove that at least 9 line segments are needed.

If point $A$ is not an endpoint of any line segment, then due to (1), the other 6 points must connect at least $C_{6}^{2}>9$ line segments.

If point $A$ is the endpoint of only 1 line segment, then due to (1), the 5 points not connected to $A$ must connect at least $C_{5}^{2}>9$ line segments.

Assume each point is the endpoint of at least two line segments. Then if point $A$ is the endpoint of only two line segments $A B, A C$, by (1), the 4 points not connected to $A$ must connect at least $C_{4}^{2}=6$ line segments. At least one more line segment must be drawn from point $B$, so in this case, there are at least $2+6+1=9$ line segments.

If each point is the endpoint of at least 3 line segments, then there must be more than $\left[\frac{3 \times 7}{2}\right]=10$ line segments.