We consider the problem of computing k∈ N internally vertex-disjoint paths between special vertex pairs of simple connected graphs. For general vertex pairs, the best deterministic time bound is, since 42 years, O(min{k,n}m) for each pair by using traditional flow-based methods. The restriction of our vertex pairs comes from the machinery of maximal adjacency orderings (MAOs). Henzinger showed for every MAO and every 1 ≤ k≤ δ (where δ is the minimum degree of the graph) the existence of k internally vertex-disjoint paths between every pair of the last δ- k+ 2 vertices of this MAO. Later, Nagamochi generalized this result by using the machinery of mixed connectivity. Both results are however inherently non-constructive. We present the first algorithm that computes these k internally vertex-disjoint paths in linear time O(n+ m) , which improves the previously best time O(min{k,n}m). Due to the linear running time, this algorithm is suitable for large graphs. The algorithm is simple, works directly on the MAO structure, and completes a long history of purely existential proofs with a constructive method. We extend our algorithm to compute several other path systems and discuss its impact for certifying algorithms.
