The Hierarchy of Probe Interval, Tolerance, and Interval k-Graphs

Abstract

We introduce a series of generalizations of probe interval graphs called t-probe interval graphs, (a probe interval graph is a 1-probe interval graph) and show, via a method similar to graph homomorphism, that each class, including the class of probe interval graphs, is contained in the class of interval k-graphs. Any probe interval graph is clearly a tolerance graph, but for some t\u3e1 this relationship fails. We wish to determine this t. Also, the interval k-graphs whose complement describes a poset are believed to have a nice characterication via forbidden subgraphs, and we give the conjecture here, and a new description of these interval k-graphs that is similar to the salient property of function graphs