The random cluster model on a general innnite graph with bounded degree wired at innnity is considered and a \ghost vertex" method is introduced in order to explicitly construct random cluster measures satisfying the Dobrushin-Lanford-Ruelle condition for q 1. It is proved that on a regular nonamenable graph there is a q 0 such that for q q 0 there is a phase transition for an entire interval o...

Parallel algorithms for recognizing and representing interval orders are proposed for differ. ent models of parallel random access machines (PRAM). The algorithms accept as input a transitively~losed directed graph with N nodes and M edges. They run in time O0og N) with O(N + M~ processors and O{N + M} space and in co , tam time with O(N z) proces~rs and O{N ~) space depending on the data struc...

We study the problem of adding edges to an arbitrary graph so that the resulting graph is an interval graph. Our objective is to add an inclusion minimal set of edges, which means that no proper subset of the added edges can result in an interval graph when added to the original graph. This problem is closely related to the problem of adding an inclusion minimal set of edges to a graph to obtai...

The minimal interval completion problem consists in adding edges to an arbitrary graph so that the resulting graph is an interval graph; the objective is to add an inclusion minimal set of edges, which means that no proper subset of the added edges can result in an interval graph when added to the original graph. We give an O(n) time algorithm to obtain a minimal interval completion of an