Dress, Andreas and Huber, Katharina and Steel, Mike - A matroid associated with a phylogenetic tree

dmtcs:2078 - Discrete Mathematics & Theoretical Computer Science, June 20, 2014, Vol. 16 no. 2
A matroid associated with a phylogenetic tree

Authors: Dress, Andreas and Huber, Katharina and Steel, Mike

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.


Source : oai:HAL:hal-01185618v1
Volume: Vol. 16 no. 2
Section: PRIMA 2013
Published on: June 20, 2014
Submitted on: December 19, 2013
Keywords: Discrete mathematics,matroid,Trees,combinatorics,linear algebra,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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