Hypersequent calculi, that are a generalization of sequent calculi, have been studied for Gödel-Dummett logics LC and LCn. In this paper we propose a new characterization of validity in these logics from the construction of particular bi-colored graphs associated to hypersequents and the search of specific chains in such graphs. It leads to other contributions that are a new hypersequent calculus and a related tableau system for LCn.We mainly study the class of so-called basic hypersequents and then we generalize our approach to hypersequents.