Extending partial representations of interval graphs

Abstract

We initiate the study of the computational complexity of the question of extending partial representations of geometric intersection graphs. In this paper we consider classes of interval graphs - given a collection of real intervals that forms an intersection representation of an induced subgraph of an input graph, is it possible to add intervals to achieve an intersection representation of the entire graph? We present an O(n2) time algorithm that solves this problem and constructs a representation if one exists. Our algorithm can also be used to list all nonisomorphic extensions with O(n2) delay. Although the classes of proper and unit interval graphs coincide, the partial representation extension problems differ on them. We present an O(mn) time decision algorithm for partial representation extension of proper interval graphs, but for unit interval graphs the complexity remains open. Finally we show how our methods can be used for solving the problem of simultaneous interval representations. We prove that this problem is fixed-paramater tractable with the size of the common intersection of the input graphs being the parameter. © 2011 Springer-Verlag.