Tolerance competition graphs

AbstractThe ø-tolerance competition graph is introduced as a generalization of the p-competition graphs defined by Kim, McKee, McMorris, and Roberts. Let ø be a symmetric function defined from N × N into N, where N denotes the nonnegative integers. G = (V, E) is a ø-tolerance competition graph if there is a directed graph D = (V, A) and an assignment of a nonnegative integer ti to each vertex vi ∈ V such that, for i ≠ j, vivj ∈ E(G) if and only if |O(vi) ∩ O(vj)| ≥ ø(ti, tj), where O(x) = {y : xy ∈ A}. A general characterization of ø-tolerance competition graphs is given, and specific results are obtained when ø is the minimum, maximum, and sum functions