On shortest disjoint paths in planar graphs

For a graph GG and a collection of vertex pairs {(s1,t1),…,(sk,tk)}{(s1,t1),…,(sk,tk)}, the kk disjoint paths problem is to find kk vertex-disjoint paths P1,…,PkP1,…,Pk, where PiPi is a path from sisi to titi for each i=1,…,ki=1,…,k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths PiPi have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k=2,3k=2,3.Min-Sum: We extend recent results by Colin de Verdière and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time.Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3.