2-Tree probe interval graphs have a large obstruction set

Abstract

Probe interval graphs ( PIGs) are used as a generalization of interval graphs in physical mapping of DNA. G = ( V , E ) is a probe interval graph ( PIG) with respect to a partition ( P , N ) of V if vertices of G 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 P; vertices belonging to P are called probes and vertices belonging to N 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 2-tree PIGs; in particular, we show that there are at least 62 minimal forbidden induced subgraphs for 2-tree PIGs.