Linear algorithms to recognize interval graphs and test for the consecutive ones property

Abstract

A matrix of zeroes and ones is said to have the consecutive ones property if there is a permutation of its rows such that the ones in each column appear consecutively. This paper develops a data structure which may be used to test a matrix for the consecutive ones property, and produce the desired permutation of the rows, in linear time. One application of the consecutive ones property is in recognizing interval graphs. A graph is an interval graph if there exists a i-i correspondence between its vertices and a set of intervals on the real line such that two vertices are adjacent if and only if the corresponding intervals have a nonempty intersection. Fulkerson and Gross have characterized interval graphs as those for which the clique versus vertex incidence matrix has the consecutive ones property. In testing this particular matrix for the consecutive ones property we may process the columns in a special order to simDlifv the algorithm. This yields the interval graph recognition algorithm which is presented in section 2; section 3 indicates how this algorithm may be extended to the general consecutive ones problem. A final section of the paper gives a number of further applications of the ideas developed in the earlier sections. These applications include linear algorithms to a) recognize unit interval graphs, b) test for the circular ones property, c) recognize planar graphs, d) count the number of distinct models of an interval graph (assuming that an arithmetic operation can be done in constant time), and e) determine whether two interval graphs are isomorphic.