Abstract
Special issue PRIMA 2013 A (pseudo-)metric D on a finite set X is said to be a \textquotelefttree metric\textquoteright if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y ∈X, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is – up to canonical isomorphism – uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree\textquoterights edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (\textquoteleftlasso\textquoteright) these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the \textquotelefttight edge-weight lassos\textquoteright for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T.
Highlights
Given any finite tree T without vertices of degree 2, there is an associated matroid M(T ) having ground set where X is the set of leaves of T
L := L1 L2 = x1x2, x2x3, x3x1, x4x5, x5x6, x6x4 is an independent subset of in M(T ). This implies that a binary X-tree T for which M(T ) is a binary matroid must be a caterpillar tree
An arbitrary X-tree T for which M(T ) is a binary matroid must be either a star tree with at most five leaves or an X-tree for which – as in the case of the caterpillar trees – two interior vertices u and v of T exist for which the path from u to v in T passes every interior vertex of T
Summary
Given any finite tree T without vertices of degree 2, there is an associated matroid M(T ) having ground set where X is the set of leaves of T. The motivation for studying this matroid is its relevance to the problem of uniquely reconstructing an edge-weighted tree from its topology and just some of the leaf-to-leaf distances in that tree. Mike Steel fixed (un-weighted) tree T and the set of minimal subsets L of for which the leaf-to-leaf distances between all x, y ∈ X with {x, y} ∈ L relative to some edge-weighting ω of T suffice to determine all the other distances relative to ω and, the edge-weighting ω. These subsets form the bases of the matroid M(T ) that will be studied here. We provide a number of remarks, observations, and questions for possible further study
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