Competition polysemy

Following a suggestion of Tanenbaum (Electron. J. Combin. 7 (2000) R43) we introduce the notion of competition polysemic pairs of graphs. A pair of (simple) graphs (G"1,G"2) on the same set of vertices V is called competition polysemic, if there exists a digraph D=(V,A) such that for all u,v@?V with u v, uv is an edge of G"1 if and only if there is some w@?V such that uw->@?A and vw->@?A and uv is an edge of G"2 if and only if there is some w@?V such that wu->@?A and wv->@?A. Our main results are a characterization of competition polysemic pairs (G"1,G"2) in terms of edge clique covers of G"1 and G"2 and a characterization of the connected graphs G for which there exists a tree T such that (G,T) is competition polysemic.