Tolerance Competition Graphs

The phi-tolerance competition graph is introduced as a generalization of the p-competition graphs defined by Kim, McKee, McMorris, and Roberts. Let phi be a symmetric function defined from N x N into N, where N denotes the nonnegative integers. G = (V, E) is a phi-tolerance competition graph if there is a directed graph D = (V, A) and an assignment of a nonnegative integer t(i) to each vertex v(i) is an element of V such that, for i not equal j, v(i)v(j) is an element of E(G) if and only if \O(v(i)) boolean AND O(v(j))\ greater than or equal to phi(t(i),t(j)), where O(x) = {y : xy is an element of A}. A general characterization of phi-tolerance competition graphs is given, and specific results are obtained when phi is the minimum, maximum, and sum functions