This work has two targets. First, we gather data on the board of directors of companies traded on the Italian Stock Market, and we analyze the structure of the related network. Second, we measure the overlap among the network of the board of directors and the shareholding network, and we evaluate the relevance of the board of directors on control, effective control, and cross-ownership relationships. In fact, shareholding data, that we have already analyzed in a previous work, capture a part only of the dependence structure, while the companies that are on the market since many years interact most through social means and common directors. Thus, such approach is a first step towards the merge of social interaction information into the analysis of ties of companies. Although we are aware that a complete analysis of social interaction and families would reveal extra complex interactions, we show how public available information changes the set of controller and controlled companies. The results point out a cause of structural risk of the market that is not considered by usual econometric models for time series of returns, and it provides a basis for the development of new indicators for risk that overcome the standard approaches most based on the time series analysis of raw return time series. Methods used in the present analysis involve statistical analyses most proper of the field of complex networks and graph flow analysis typical of operations research approach. The dataset reports the board of directors (CONSOB database), and the shareholdings of nearly 300 companies traded on the Italian Stock Market. Data, adjourned at May 2008, were retrieved through the AIDA database, integrated by the Bureau van Dijk databases BANKSCOPE, ISIS, and cross-validated through CONSOB and MEDIOBANCA reports.