Edge-disjoint paths in a grid bounded by two nested rectangles

This paper presents an algorithm for finding edge-disjoint paths in a given plane grid bounded by two nested rectangles. A pair of vertices on the boundary of the same rectangle are designated as terminals for each of the paths. The number of terminals lying on each boundary vertex is determined by the degree of the boundary vertex. Every vertex of degree 2 has either 0 or 2 terminals lying on it, every vertex of degree 3 has exactly one terminal, and every vertex of degree 4 has no terminal. If there are n vertices in the grid and bt vertices on the outer rectangle, then the algorithm decides in O(bt) time whether there exist edge-disjoint paths, and actually finds the paths in O(n) time if there exist.