Recognition Algorithm for Probe Interval 2-Trees

Recognition of probe interval graphs has been studied extensively. Recognition algorithms of probe interval graphs can be broken down into two types of problems: partitioned and non-partitioned. A partitioned recognition algorithm includes the probe and nonprobe partition of the vertices as part of the input, where a non-partitioned algorithm does not include the partition. Partitioned probe interval graphs can be recognized in linear-time in the edges, whereas non-partitioned probe interval graphs can be recognized in polynomial-time. Here we present a non-partitioned recognition algorithm for 2-trees, an extension of trees, that are probe interval graphs. We show that this algorithm runs in O(m) time, where m is the number of edges of a 2-tree. Currently there is no algorithm that performs as well for this problem

2-tree probe interval graphs have a large obstruction set

Probe interval graphs are used as a generalization of interval graphs in physical mapping of DNA. is a probe interval graph (PIG) with respect to a partition ¥���¨��� � of ¦ if vertices of ¢ correspond to intervals on a real line and two vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is in �; vertices belonging to � are called probes and vertices belonging to � are called non-probes. One common approach to studying the structure of a new family of graphs is to determine if there is a concise family of forbidden induced subgraphs. It has been shown that there are two forbidden induced subgraphs that characterize tree PIGs. In this paper we show that having a concise forbidden induced subgraph characterization does not extend to �-tree PIGs; in particular we show that there are at least sixty-two minimal forbidden induced subgraphs for �-tree PIGs

Probe split graphs

Graphs and Algorithm

A Characterization of 2-Tree Probe Interval Graphs

A graph is a probe interval graph if its vertices correspond to some set of intervals of the real line and can be partitioned into sets P and N so that vertices are adjacent if and only if their corresponding intervals intersect and at least one belongs to P. We characterize the 2-trees which are probe interval graphs and extend a list of forbidden induced subgraphs for such graphs created by Pržulj and Corneil in [2-tree probe interval graphs have a large obstruction set, Discrete Appl. Math. 150 (2005) 216-231