Disjoint unions of complete minors

The Complexity of Unions of Disjoint Sets

This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions o...

Disjoint paths in unions of tournaments

Given k pairs of vertices (si, ti) (1 ≤ i ≤ k) of a digraph G, how can we test whether there exist vertex-disjoint directed paths from si to ti for 1 ≤ i ≤ k? This is NP-complete in general digraphs, even for k = 2 [4], but in [3] we proved that for all fixed k, there is a polynomial-time algorithm to solve the problem if G is a tournament (or more generally, a semicomplete digraph). Here we pr...

Canonization for Disjoint Unions of Theories

If there exist efficient procedures (canonizers) for reducing terms of two first-order theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, we prove that algorithms for solving equations in the two theories (solvers) cannot be combine...

Canonization for Disjoint Unions of Theories

Disjoint Unions of Topological Spaces and Choice

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice. Introduction and Terminology This is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also [vd], [gt], [wgt], and [hkrr]. Our primary concern will be the use of the axiom of choice in proving properties of disjoint unions ...