Representations of Graphs by Means of Products and Their Complexity

I. A graph G is a pair (V,E) where V is a finite set and E is either a set of ordered pairs from V (i.e. E ~V × V) or a set of unordered pairs from V (i.e. E ~ [V3 2 ) . In the former case G is called an oriented graph in the latter an unoriented graph . Elements of the set E are called edges. An edge e g E is indicated by [x,y] = e where the bracketing means consistently in the whole formula either (x,y) (an arrow an ordered edge) or ~x,y} Ca link an unordered edge) .