On the Encoding of Relations by Graphs

Encodings of relations by graphs are viewed as 1–1 mappings from relations to directed graphs. A measure called the size of such encodings is defined. A subclass of encodings called translations is introduced. These are essentially encodings decodable by a first order formula. Lower bounds on their sizes are proven. We present a translation whose size is asymptotically equal to the lower bound, differing from it only in low order terms.A closely related notion of perfect encoding is defined as being a 1–1 and onto mapping, that also has the interesting property that every edge in an encoding graph corresponds directly to a tuple in the original relation, and vice-versa. A perfect encoding is constructed and using it, a classical result about random graphs is converted into a result about random relations. It is believed that many other results can be similarly converted using these notions.Generalizations of this work to encodings of relations by relations of lower degree are described, and analogous methods are shown to work.