Edge-disjoint homotopic paths in a planar graph with one hole

AbstractWe prove the following theorem, conjectured by K. Mehlhorn: Let G = (V, E) be a planar graph, embedded in the plane C. Let O denote the interior of the unbounded face, and let I be the interior of some fixed bounded face. Let C1, …, Ck be curves in Cß(I⌣O), with end points in V⌢bd(I⌣O), so that for each vertex v of G the degree of v in G has the same parity as the number of curves Ci beginning or ending in v (counting a curve beginning and ending in v for two). Then there exist pairwise edge-disjoint paths P1, …, Pk in G so that Pi is homotopic to Ci in the space Cß(I⌣O) for i = 1, …, k, if and only if for each dual walk Q from {I, O} to {I, O} the number of edges in Q is not smaller than the number of times Q necessarily intersects the curves Ci. The theorem generalizes a theorem of Okamura and Seymour. We demonstrate how a polynomial-time algorithm finding the paths can be derived

Shortest disjoint paths on a grid

The well-known k-disjoint paths problem involves finding pairwise vertex-disjoint paths between k specified pairs of vertices within a given graph if they exist. In the shortest k-disjoint paths problem one looks for such paths of minimum total length. Despite nearly 50 years of active research on the k-disjoint paths problem, many open problems and complexity gaps still persist. A particularly well-defined scenario, inspired by VLSI design, focuses on infinite rectangular grids where the terminals are placed at arbitrary grid points. While the decision problem in this context remains NP-hard, no prior research has provided any positive results for the optimization version. The main result of this paper is a fixed-parameter tractable (FPT) algorithm for this scenario. It is important to stress that this is the first result achieving the FPT complexity of the shortest disjoint paths problem in any, even very restricted classes of graphs where we do not put any restriction on the placements of the terminals

On local routing of two-terminal nets

SIGLETIB Hannover: RO 1829(1986,3) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman