### Abstract:

Given k terminal pairs (s₁,t₁),(s₂,t₂),..., (s[subscript k],t[subscript k]) in an edge-weighted graph G, the k Shortest Vertex-Disjoint Paths problem is to find a collection P₁, P₂,..., P[subscript k] of vertex-disjoint paths with minimum total length, where P[subscript i] is an s[subscript i]-to-t[subscript i] path. As a special case of the multi-commodity flow problem, computing vertex disjoint paths has found several applications, for example in VLSI design, or network routing. In this thesis we describe a Structural Theorem for a special case of the Shortest Vertex-Disjoint Paths problem in undirected planar graphs where the terminal vertices are on the boundary of the outer face. At a high level, our Structural Theorem guarantees that the i[superscript th] path of the k Shortest Vertex-Disjoint paths does not cross j[superscript th] (j ≠ i) path of the k-1 Vertex-Disjoint Paths problem.