A linear-time algorithm for finding a paired 2-disjoint path cover in the cube of a connected graph

For a connected graph G = (V (G), E(G)) and two disjoint subsets of V (G) A = {α1, . . . , αk} and B = {β1, . . . , βk}, a paired (many-to-many) k-disjoint path cover of G joining A and B is a vertex-disjoint path cover {P1, . . . , Pk} such that Pi is a path from αi to βi for 1 ≤ i ≤ k. In the recent paper [Disjoint Path Covers in Cubes of Connected Graphs, Discrete Mathematics 325 (2014) 65–73], Park and Ihm presented a necessary and sufficient condition for a paired 2-disjoint path cover joining two vertex sets to exist in the cube of a connected graph. In this paper, we propose an O(|V (G)| + |E(G)|)time algorithm that actually finds such a paired 2-disjoint path cover. In particular, we show that, in order to build a desired disjoint path cover, it is sufficient to consider only the edges of a carefully selected spanning tree of the graph and at most one additional edge not in the tree, which allows an efficient linear-time algorithm.