Two-Phi-Tolerance Competition Graphs

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 two-phi-tolerance competition graph is a phi-tolerance competition graph in which all the t(i) are selected from a 2-set. Characterization of such graphs, and relationships between them are presented for phi equal to the minimum, maximum, and sum fractions, with emphasis on the situation in which the 2-set is {0, q}