Disjoint Paths in a Planar Graph—A General Theorem

Abstract

Let $D = ( V,A )$ be a directed planar graph, let $( r_1 ,s_1 ), \cdots , ( r_k ,s_k )$ be pairs of vertices on the boundary of the unbounded face, let $A_1 , \cdots ,A_k $ be subsets of A, and let H be a collection of unordered pairs from $\{ 1, \cdots ,k \}$. Given are necessary and sufficient conditions for the existence of a directed $r_i - s_i $ path $P_i $ in $( V,A_i )$ (for $i = 1, \cdots ,k$), such that $P_i $ and $P_j $ are vertex-disjoint whenever $\{ i, j \} \in H$.