Larry J. Langley

Tolerance orders and bipartite unit tolerance graphs

Abstract We construct all six-element orders which are not 50%-tolerance orders. We show that a width-two order is a 50% tolerance order if and only if no restriction of the order to a six-element set is isomorphic to one of these six-element orders. This yields a corresponding characterization of bipartite 50%-tolerance graphs. Since an order (graph) has a 50% tolerance representation if and only if it has a unit tolerance representation, our results apply to unit tolerance orders (graphs) as well.

Probe Interval Orders

A probe interval graph is a graph with vertex partition P ∪ N and to each vertex v there corresponds an interval Iv such that vertices are adjacent if and only if their corresponding intervals intersect and at least one of the vertices belongs to P . If a graph has a transitive orientation on its complement, it is a cocomparability graph, and we can think of it as the incomparability graph of the order given by a transitive orientation of its complement. When the vertices of N have a proper representation (no interval contains another properly), a natural transitive orientation of the complement occurs. We call the order that arises a probe interval order. We characterize which probe interval graphs yield a probe interval order by restrictions placed on {Iv : v ∈ N}, and by the nature of the partition restricted to 4-cycles in the graph. We discuss methods for recognizing cocomparability probe interval graphs, both in the partitioned and non-partitioned case.

Interval k-Graphs and Orders

An interval k-graph is the intersection graph of a family of intervals of the real line partitioned into k classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we study the cocomparability interval k-graphs; that is, the interval k-graphs whose complements have a transitive orientation and are therefore the incomparability graphs of strict partial orders. For brevity we call these orders interval k-orders. We characterize the kind of interval representations a cocomparability interval k-graph must have, and identify the structure that guarantees an order is an interval k-order. The case k = 2 is peculiar: cocomparability interval 2-graphs (equivalently proper- or unit-interval bigraphs, bipartite permutation graphs, and complements of proper circular-arc graphs to name a few) have been characterized in many ways, but we show that analogous characterizations do not hold if k > 2. We characterize the cocomparability interval 3-graphs via one forbidden subgraph and hence interval 3-orders via one forbidden suborder.

Kings and Heirs

In 1980, Maurer coined the phrase king when describing any vertex of a tournament that could reach every other vertex in two or fewer steps. A ( 2 , 2 ) -domination graph of a digraph D , d o m 2 , 2 ( D ) , has vertex set V ( D ) , the vertices of D , and edge u v whenever u and v each reach all other vertices of D in two or fewer steps. In this special case of the ( i , j ) -domination graph, we see that Maurers theorem plays an important role in establishing which vertices form the kings that create some of the edges in d o m 2 , 2 ( D ) . But of even more interest is that we are able to use the theorem to determine which other vertices, when paired with a king, form an edge in d o m 2 , 2 ( D ) . These vertices are referred to as heirs. Using kings and heirs, we are able to completely characterize the ( 2 , 2 ) -domination graphs of tournaments.