A characterization of competition graphs of arbitrary digraphs

Let D = (K/;A) be a digraph which may have loops, i.e., arcs of the form (x,x). The competition graph of D is the undirected graph G obtained as follows. V(G) = I/ and {x, y> EE(G) if and only if for some z E V, arcs (x, z) and (y, z) are in A. Competition graphs were introduced by Cohen [1968] in connection with problems of ecology. Cohen was primarily interested in the case where D is an acyclic digraph. This case is also discussed by Cohen [1977, 19781, Dutton and Brigham [1983], Lundgren and Maybee [ 19831, Opsut [1982], and Roberts [1978a,b]. Dutton and Brigham [ 19831 introduce the study of competition graphs of digraphs D which may not be acyclic. A notion similar to that of competition graph arises when D is a bipartite digraph with all arcs heading from a set S to a set T. The subgraph of the competition graph which is generated by vertices in S is the confusion graph which arises in the study of communication over noisy channels (Shannon [1956], Roberts [1978b]). Here, the set S is a transmission alphabet and the set T is a receiving alphabet, and an arc from s to t means that s can be received as t. A similar construction gives rise to the conflict graph which is relevant to the problem of assigning channels to radio or television transmitters (Cozzens and Roberts [1982], Hale (19801, Opsut and Roberts [1981]). Here, the set S is taken to be a set of transmitters and the set T to be a set of locations, and an arc from s to t means that transmitter s can be heard at location 1. Dutton and Brigham characterize the competition graphs arising from digraphs D which may have cycles and which also may have loops. We shall obtain a similar characterization in the case that there are no loops. Let m(G) be the smallest number of cliques (not necessarily maximal) of G which cover all edges of G. Orlin [I9771 and Kou, et al. [1978] prove that computation of m(G) is an NP-complete problem. Dutton and Brigham [1983] prove the following theorem.