The unilateral 2-linkage problem
We study a new variant of the classical linkage problem for digraphs, which we call {\bf unilateral linkages.} A digraph $D$ is {\bf unilaterally $\bm{k}$-linked} if it satisfies that for every choice of $2k$ distinct vertices $s_1,s_2,\ldots{},s_k,t_1,t_2,\ldots{},t_k$ there are vertex disjoint directed paths $P_1,\ldots{},P_k$ in $D$ such that $P_i$ is either an $(s_i,t_i)$-path or a $(t_i,s_i)$-path for each $i\in [k]$. This is a relaxation of the property of $D$ being $k$-linked, where we insist that each $P_i$ must go from $s_i$ to $t_i$. Clearly every $k$-linked digraph is also unilaterally $k$-linked. It is easy to see that the unilateral linkage problem is NP-complete for general digraphs and that it is trivial for semicomplete digraphs (digraphs with no pair of non-adjacent vertices). We show how Thomassen's construction of highly connected digraphs which are not 2-linked can be modified to show that there is no connectivity which guarantees that a digraph is unilaterally 2-linked. We show that every 4-strong split digraph and every 5-strong semicomplete multipartite digraph is unilaterally 2-linked. Both bounds are best possible. A digraph is {\bf co-bipartite} if its vertex set can be covered by two induced semicomplete digraphs. We prove that every 6-strong co-bipartite digraph is unilaterally 2-linked. Here we do not know whether the correct answer is 5. We also prove that every 16-strong digraph with independence number at most 2 is unilaterally 2-linked.