A New Test for Interval Graphs

An interval graph is the intersection graph of a collection of intervals. Interval graphs are a special class of chordal graphs. This class of graphs has a wide range of applications. Several linear time algorithms have been designed to recognize interval graphs. Booth & Lueker first used PQ-trees to recognize interval graphs in linear time. However, the data manipulation of PQ-trees is rather involved and the complexity analysis is also quite tricky. Korte and Möhring simplified the operations on a PQ-tree using an incremental algorithm. Hsu and Ma gave a simpler decomposition algorithm without using PQ-trees. All of these algorithms rely on the following fact : a graph is an interval graph iff there exists a linear order of its maximal cliques such that for each vertex v, all maximal cliques containing v are consecutive. Thus, the precomputation of all maximal cliques is required for these algorithms. Based on graph decomposition, we give a much simpler recognition algorithm in this paper which directly places the intervals without precomputing all maximal cliques. A linear time isomorphism algorithm can be easily derived as a by-product. Another advantage of our approach is that it can be used to develop an O(nlog n) on-line recognition algorithm for interval graphs.