We consider a graph representation for a paragraph of text. It widely uses linguistic theories of discourse to extend the set of edges between vertices corresponding to words. Parse thickets is a set of syntactic parse trees augmented by a number of inter-sentence coreference links and links based on Speech Act and Rhetoric Structures Theories. Similarity of parse thickets is defined by means of intersection operation taking common parts of the thickets. Several approaches to computing intersection of parse thickets are proposed and compared. Projections as approximation means are considered.