We consider the problem of computing k is an element of N internally vertex-disjointpaths between special vertex pairs of simple connected graphs. For general vertex pairs, the best deterministic time bound is, since...
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We consider the problem of computing k is an element of N internally vertex-disjointpaths 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 <=delta (where delta is the minimum degree of the graph) the existence of k internally vertex-disjointpaths between every pair of the last delta-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-disjointpaths 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.
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