Interval k-Graphs and Orders

Abstract

An interval k-graph is the intersection graph of a family I of intervals of the real line partitioned into at most k classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we discuss the interval k-graphs that are the incomparability graphs of orders; i.e., cocomparability interval k-graphs or interval k-orders. Interval 2-orders have been characterized in many ways, but we show that analogous characterizations do not carry over to interval k-orders, for k > 2. We describe the structure of interval k-orders, for any k, characterize the interval 3-orders (cocomparability interval 3-graphs) via one forbidden suborder (subgraph), and state a conjecture for interval k-orders (any k) that would characterize them via two forbidden suborders.

$L(p,q)$-labeling of graphs with interval representations

“…The family of interval k-graphs is a relatively new class, firstly introduced in [5] as a generalization of (probe) interval graphs and interval bigraphs. Brown [3] provides a characterization of interval k-graphs in terms of consecutive ordering of its complete r-partite subgraphs.…”

“…Brown [3] provides a characterization of interval k-graphs in terms of consecutive ordering of its complete r-partite subgraphs. As there is no further characterization known for this class, the recent works concentrate on the possible characterization of its subclasses such as cocomparability interval k-graphs [4] and AT-free interval k-graphs [12].…”

Veto Interval Graphs and Variations

“…Interval graphs have been extensively studied and characterized, and fast algorithms for finding the clique number, chromatic number, and other graph parameters have been developed [9]. Furthermore, many variations of interval graphs, including interval p-graphs, interval digraphs, circular arc graphs, and probe interval graphs, have been introduced and investigated [4,5,11,13].…”