Edge‐arc‐disjoint paths in semicomplete mixed graphs

Abstract The so‐called weak‐2‐linkage problem asks for a given digraph and distinct vertices of whether has arc‐disjoint paths so that is an ‐path for . This problem is NP‐complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge‐disjoint and arc‐disjoint in semicomplete mixed graphs, that is, a mixed graph in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.