\(p\)-competition graphs (Q1805319)

The \(p\)-competition graph of a digraph \(D= (V, A)\) is the graph with vertex set \(V\) and an edge between \(x\) and \(y\) whenever there exist distinct vertices \(v_ 1, v_ 2,\dots, v_ p\) and arcs \((x, v_ i)\), \((y,v_ i)\) in \(D\) for each \(i\leq p\). In case \(p= 1\) the concept coincides with the competition graph which has received a great deal of attention in the literature since being introduced by J. E. Cohen in 1968, see, e.g., \textit{R. D. Dutton} and \textit{R. C. Brigham} [A characterization of competition graphs, Discrete Appl. Math. 6, 315-317 (1983; Zbl 0521.05057)]. In the present paper, the authors study properties of \(p\)-competition graphs, obtaining, where possible, analogues of results about ordinary competition graphs.