A Linear-Time Algorithm for Edge-Disjoint Paths in Planar Graphs

TLDR: A new approach to the problem of finding edge-disjoint paths in a planar, undirected graph with “classical” case where an instance must additionally fulfill the so-called evenness-condition is introduced.

Abstract: In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph s.t. each path connects two specified vertices on the outer face boundary. We will focus on the “classical” case where an instance must additionally fulfill the so-called evenness-condition. The fastest algorithm for this problem known from the literature requires \(\mathcal{O}\left( {n^{{5 \mathord{\left/{\vphantom {5 3}} \right.\kern-\nulldelimiterspace} 3}} \left( {\log \log n} \right)^{{1 \mathord{\left/{\vphantom {1 3}} \right.\kern-\nulldelimiterspace} 3}} } \right)\) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which yields an \(\mathcal{O}\left( n \right)\) algorithm.