Computing bounded-degree phylogenetic roots of disconnected graphs

The Phylogenetic kth Root Problem (PRk) is the problem of finding a (phylogenetic) tree T from a given graph G=(V,E) such that (1) T has no degree-2 internal nodes, (2) the external nodes (i.e., leaves) of T are exactly the elements of V, and (3) (u,v)E if and only if the distance between u and v in tree T is at most k, where k is some fixed threshold k. Such a tree T, if exists, is called a phylogenetic kth root of graph G. The computational complexity of PRk is open, except for k4. Recently, Chen et al. investigated PRk under a natural restriction that the maximum degree of the phylogenetic root is bounded from above by a constant. They presented a linear-time algorithm that determines if a given connected G has such a phylogenetic kth root, and if so, demonstrates one. In this paper, we supplement their work by presenting a linear-time algorithm for disconnected graphs.