In recent works, we have proposed a graphical model to represent linguistic preferences called LCP-nets. LCP-nets have been implemented and used in a specific use case of industrial engineering. In this paper, we consolidate this contribution in formalizing it through a set of notations and computation rules in order to guarantee its durability and its reusability to other multi-criteria decision contexts. The paper formalizes the LCP-net structure, semantics, and validity. It also formalizes the dominance testing and optimization queries (for a discretized version of the problem in this latter case), in the line of previous CP-nets models.