Abstract
It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset $\mathcal{L}$ of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in $\mathcal{L}$. It is known that any set ${\mathcal L}$ with this property for a tree in which all interior vertices have degree 3 must form a cover for $T$ - that is, for each interior vertex $v$ of $T$, ${\mathcal L}$ must contain a pair of leaves from each pair of the three components of $T-v$. Here we provide a partial converse of this result by showing that if a set ${\mathcal L}$ of leaf pairs forms a cover of a certain type for such a tree $T$ then $T$ and its edge weights can be uniquely determined from the distances between the pairs of leaves in ${\mathcal L}$. Moreover, there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning 'triplet covers', and is relevant to a problem arising in evolutionary genomics.
Highlights
Any tree T with positively weighted edges, induces a metric d on the set of leaves by considering the weighted path distance in T between each pair of leaves
It is known that any set L with this property for a tree in which all interior vertices have degree 3 must form a cover for T – that is, for each interior vertex v of T, L must contain a pair of leaves from each pair of the three components of T − v
We provide a partial converse of this result by showing that if a set L of leaf pairs forms a cover of a certain type for such a tree T T and its edge weights can be uniquely determined from the distances between the pairs of leaves in L
Summary
Any tree T with positively weighted edges, induces a metric d on the set of leaves by considering the weighted path distance in T between each pair of leaves. Provided T has no vertices of degree 2, and that we ignore the labeling of interior vertices, both T and its edge weights are uniquely determined by the metric d This uniqueness the electronic journal of combinatorics 21(2) (2014), #P2.15 result has been known since the 1960s and fast algorithms exist for reconstructing both the tree and its edge weights from d (for further background the interested reader may consult [1] and [10] and the references therein). The uniqueness result and the algorithms are important in evolutionary biology for reconstructing an evolutionary tree of species from genetic data [6] In this setting one frequently may not have d-values available for all pairs of species, due to the patchy nature of genomic coverage [9]. We conclude by providing a proof that a polynomial-time algorithm will reconstruct a tree and its edge weights for any set of leaf pairs that contains a stable triplet cover (or more generally a shellable subset). Our result answers a special case of the question posed at the end of [4] of whether every ‘triplet cover’ of a fully-resolved tree determines the tree and its edge weights
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