The design of future multi-standard systems is very challenging. Flexible architectures exploiting processing commonalities of the di erent set of standards cohabiting in the device o er promising solutions. In this paper, graph theory aspects are introduced with the stress on the notion of directed hypergraphs which will help in showing how the graph structure of the multi-standard system can be viewed as a clear theoretical directed hypergraph. This graph description provides all the alternatives capable of implementing the design. However, a cost function needs to be de ned to calculate the cost of each possible option chosen. Thus, we explain the sound concept of the computation process of the cost function introduced in previous papers, before transforming it into a clear formal equation with the aid of various de nitions and notations of directed hypergraphs.