A Linear Time Algorithm for the Arc Disjoint Menger Problem in Planar Directed Graphs1

Given a graph G = (V, E) and two vertices s, t ∈ V , s 6= t , the Menger problem is to find a maximum number of disjoint paths connecting s and t . Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or arc disjoint Menger problem, respectively. For planar graphs the edge disjoint Menger problem has been solved to optimality [W2], while the fastest algorithm for the arc disjoint version is Weihe’s general maximum flow algorithm for planar networks [W1], which has running timeO(|V | log|V |). Here we present a linear time, i.e., asymptotically optimal, algorithm for the arc disjoint version in planar directed graphs.