A disjoint unions theorem for trees

The Ramsey numbers for disjoint unions of trees

For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number R(∪G,H), where G is a tree and H is a wheel Wm or a complete graph Km. We show that if n ≥ 3, then R(kSn, W4) = (k+1)n for k ≥ 2, even n and R(kSn,W4) = (k+1)n−1 for k ≥ 1...

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 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...

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...