The problem of deciding whether a join dependency [R] and a set F of functional dependencies logically imply an embedded join dependency [S] is known to be NP-complete. It is shown that if the set F of functional dependencies is required to be embedded in R, the problem can be decided in polynomial time. The problem is approached by introducing agreement graphs, a type of graph structure which helps expose the combinatorial structure of dependency implication problems. Agreement graphs provide an alternative formalism to tableaus and extend the application of graph and hypergraph theory in relational database research;Agreement graphs are also given a more abstract definition and are used to define agreement graph dependencies (AGDs). It is shown that AGDs are equivalent to Fagin's (unirelational) embedded implicational dependencies. A decision method is given for the AGD implication problem. Although the implication problem for AGDs is undecidable, the decision method works in many cases and lends insight into dependency implication. A number of properties of agreement graph dependencies are given and directions for future research are suggested.