A Characterization of 2-Tree Proper Interval 3-Graphs

Abstract

An intervalp-graph is the intersection graph of a collection of intervals which have been colored withpdifferent colors with edges corresponding to nonempty intersection of intervals from different color classes. We characterize the class of 2-trees which are interval 3-graphs via a list of three graphs and three infinite families of forbidden induced subgraphs.

Interval k-Graphs and Orders

“…Although the classes of unit interval k-graphs and proper interval k-graphs are identical, see [3], the analogues of statements in Theorem 2.1 extend no further for proper interval k-graphs with k > 2. In this section we show that the statements 3, 4, 5, 8, 9, 10, and 11 of Theorem 2.1 do not necessarily hold for a proper interval k-graph, k > 2.…”

“…The graph in Figure 5 is not a unit or proper interval k-graph (straightforward to verify, or see [3] or [6]), but is labeled in accord with Theorem 3.1. The next theorem follows from a result of Corneil and others [9] and since proper interval k-graphs are asteroidal triple free [6], but we give a short proof following from the ordering of Theorem 3.1.…”