Elimination Ordering Characterizations of Digraphs with Interval and Chordal Competition Graphs

Competition graphs have appeared in a variety of applications from food webs to communication networks to energy models. The competition graph, C (D), of a digraph D has the same vertex set as D and (x; y) is an edge in C (D) if and only if there is a vertex z such that (x; z) and (y; z) are arcs in D. It has been observed that most actual food webs have interval competition graphs. Subsequent research investigating this phenomenon led to the problem of characterizing acyclic digraphs which have interval competition graphs since food web models are generally acyclic. This evolved into the problem of characterizing digraphs with interval or chordal competition graphs. The problem is diicult because forbidden subgraph characterizations generally don't work. Here we use an elimination ordering approach which leads to useful characterizations which can be generalized to determine which graphs have interval or chordal squares or two-step graphs.