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

초록

For a connected graph G = (V (G), E(G)) and two disjoint subsets of V(G)A = (alpha(1), ..., alpha(k)} and B = {beta(1,) ..., beta(k)}, a paired (many-to-many) k-disjoint path cover of G joining A and B is a vertex-disjoint path cover (beta(1), ..., beta(k)} such that P-i is a path from alpha(i) to beta(i) for 1 <= i <= k. In the recent paper, Park and Ihm (2014) 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 vertical bar V(G)vertical bar + vertical bar E(G)vertical bar)-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. (C) 2016 Elsevier B.V. All rights reserved.

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