This paper analyzes structured P2P systems where peers choose both their interaction mode, i.e., how they process incoming queries, and additional contacts in the network autonomously. Since additional contacts incur additional costs, a new kind of free riding behavior, namely having only few contacts, comes into the fray. We refer to it as deliberately poor connectedness (dpc). In this paper, we show that dpc is dominant in many situations. This leads to networks with a low degree of connectivity and a higher overall forwarding load than necessary. We then propose an incentive mechanism against dpc and demonstrate its effectiveness using a formal analysis and experiments.