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.

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