Forbidden and Acceptable Line Trees

The problem of existence of a tree satisfying a given set of constraints on its vertices is considered. Specifically, given finite set of constraints P = {p1, p2, ..., pn}, we ask if there is an efficient algorithm to build a line-tree T = (V, E) such that for every constraint pi = (ab)(cd), where a, b, c, d are arbitrary elements of V , the path (ab) does not share any vertices with the path (cd) on the line-tree. After a primer on basic combinatorial mathematics and complexity theory, we make general counting observations about constraints and their recognizability. We proceed by describing an algorithm which can be used to generate an infinite list of unrecognizable constraints, thus giving the reader an intuitive notion of the complexity of our problem. Finally, we show that realizability of constraints by a line-tree is NP -complete, by doing several reductions from an already known NP-Complete problem. keywords: NP -complete, complexity theory, algorithms, graph theory