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On the complexity of the disjoint paths problem. (English) Zbl 0770.68072
Summary: We consider the disjoint paths problem. Given a graph $$G$$ and a subset $$S$$ of the edge-set of $$G$$ the problem is to decide whether there exists a family $$\mathcal F$$ of disjoint circuits in $$G$$ each containing exactly one edge of $$S$$ such that every edge in $$S$$ belongs to a circuit in $$\mathcal C$$. By a well-known theorem of P. Seymour [On odd cuts and plane multicommodity flows, Proc. London Math. Soc. (3)42, 178-192 (1981; Zbl 0447.90026)] the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphs $$G$$. We show that (assuming $$P \neq NP$$) one can drop neither planarity nor the Eulerian condition on $$G$$ without losing polynomial time solvability. We prove the NP-completeness of the planar edge-disjoint paths problem by showing the NP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assuming $$P \neq NP$$) a conjecture of A. Schrijver [Homotopic Routing Methods, in: Paths, Flows and VLSI Layout, Algorithms Comb. 9, 329-371 (1990; Zbl 0732.90087)] concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjugate of A. Frank mentioned in A. Sebö [Dual Integrality and Multicommodity Flows. Combinatorics, Colloquia Mathematica Societatis János Bolyai, 52, 453-469 (1988; Zbl 0695.90041)]. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minor-closed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show the NP-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.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 68R10 Graph theory (including graph drawing) in computer science 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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##### References:
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