Phylogenetic k-Root and Steiner k-Root

Given a graph G = (V; E) and a positive integer k, the Phylogenetic k-Root Problem asks for a (unrooted) tree T without degree-2 nodes such that its leaves are labeled by V and (u; v) 2 E if and only if dT (u; v) k. If the vertices in V are also allowed to be internal nodes in T , then we have the Steiner k-Root Problem. Moreover, if a particular subset S of V are required to be internal nodes in T , then we have the Restricted Steiner k-Root Problem. Phylogenetic k-roots and Steiner k-roots extend the standard notion of graph roots and are motivated by applications in computational biology. In this paper, we rst present O(n + e)-time algorithms to determine if a (not necessarily connected) graph G = (V; E) has an S-restricted 1-root Steiner tree for a given subset S V , and to determine if a connected graph G = (V; E) has an S-restricted 2-root Steiner tree for a given subset S V , where n = jV j and e = jEj. We then use these two algorithms as subroutines to design O(n + e)-time algorithms to determine if a given (not necessarily connected) graph G = (V; E) has a 3-root phylogeny and to determine if a given connected graph G = (V; E) has a 4-root phylogeny.