On the complexity of the disjoint paths problem

In this paper we consider the disjoint paths problem. Given a graphG and a subsetS of the edge-set ofG the problem is to decide whether there exists a family of disjoint circuits inG each containing exactly one edge ofS such that every edge inS belongs to a circuit inC. By a well-known theorem of P. Seymour the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphsG. We show that (assumingPNP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability. We prove theNP-completeness of the planar edge-disjoint paths problem by showing theNP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assumingPNP) a conjecture of A. Schrijver concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjecture of A. Frank. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minorclosed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show theNP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.Supported by Sonderforschungsbereich 303 (DFG)     

On-line multidimensional chromatography using packed capillary liquid chromatography and capillary gas chromatography

The introduction of selected fractions from a liquid chromatograph into a gas chromatograph has been described; however, analyses were performed by off-line experiments requiring collection and reinjection of the separate fractions or by on-line procedures where disadvantageously, only a fraction of the separated peak or a well resolved component in a mixture could be introduced into a gas chromatograph. This disadvantage is overcome by the apparatus and method described in this paper, which utilizes a multidimensional chromatography system employing a high efficiency, packed capillary LC column coupled on-line to a capillary gas chromatograph. The liquid chromatograph (so designed) can act as a highly efficient clean-up or chemical class fractionation step prior to introduction into the gas chromatograph, significantly reducing sample preparation times in many applications. Thus minor components in a complex matrix can be determined without prior sample clean-up, an example of which is the determination of polychlorinated biphenyls in a complex hydrocarbon matrix.