# A simple linear-time algorithm to recognize interval graphs

Specific Algorithms

First Online:

## Abstract

The fastest known algorithm for recognizing interval graphs [1] iteratively manipulates the system of all maximal cliques of the given graph in a rather complicated way in order to construct a consecutive arrangement (more precisely: a tree representation of all possible such consecutive arrangements). We present a much simpler algorithm which uses a related, but much more informative tree representation of interval graphs. This tree is constructed in an on-line fashion by adding vertices to the graph in a predefined order such that adding a vertex *u* takes *O*(|Adj(*u*)|) amortized time.

## Key words

interval graphs on-line recognition graph algorithm perfect elimination scheme modified*PQ*-tree

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]S. Booth and S. Lueker [1976]: Testing for the consecutive ones property, interval graphs, and graph planarity using
*PQ*-tree algorithms.*J. Comput. Syst. Sci. 13, 335–379*.Google Scholar - [2]D.R. Fulkerson and O.A. Gross [1965]: Incidence matrices and interval graphs.
*Pacific J. Math. 15, 835–855*.Google Scholar - [3]M.C. Golumbic [1980]: Algorithmic Graph Theory and Perfect Graphs.
*Academic Press, New York*.Google Scholar - [4]N. Korte and R.H. Möhring [1985]: Transitive orientation of graphs with side constraints.
*Proceedings “Workshop on Graphtheoretic Concepts in Computer Science 1985” (ed. H. Noltemeier) Trauner, Linz, p. 143–160*.Google Scholar - [6]D.J. Rose, R.E. Tarjan, and G.S. Lueker [1976]: Algorithmic aspects of vertex elimination of graphs.
*SIAM J. Comput. 5, 266–283*.CrossRefGoogle Scholar - [7]R.E. Tarjan Amortized computational complexity.
*SIAM J. Alg. Disc. Math., to appear*Google Scholar - [8]R.E. Tarjan and M. Yannakakis [1984]: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs.
*SIAM J. Comput. 13, 566–579*.CrossRefGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1987