   
Andreas W. M. Dress,
Katharina Huber,
Jacobus Koolen and
Vincent Moulton. Compatible decompositions and block realizations of finite metrics. In EJC, Vol. 29(7):1617-1633, 2008.
Keywords: abstract network, block realization, from distances, phylogenetic network, phylogeny, realization, reconstruction.
Note: http://www.ims.nus.edu.sg/preprints/2007-21.pdf.
Toggle abstract
"Given a metric D defined on a finite set X, we define a finite collection D of metrics on X to be a compatible decomposition of D if any two distinct metrics in D are linearly independent (considered as vectors in RX × X), D = ∑d ∈ D d holds, and there exist points x, x′ ∈ X for any two distinct metrics d, d′ in D such that d (x, y) d′ (x′, y) = 0 holds for every y ∈ X. In this paper, we show that such decompositions are in one-to-one correspondence with (isomorphism classes of) block realizations of D, that is, graph realizations G of D for which G is a block graph and for which every vertex in G not labelled by X has degree at least 3 and is a cut point of G. This generalizes a fundamental result in phylogenetic combinatorics that states that a metric D defined on X can be realized by a tree if and only if there exists a compatible decomposition D of D such that all metrics d ∈ D are split metrics, and lays the foundation for a more general theory of metric decompositions that will be explored in future papers. © 2007 Elsevier Ltd. All rights reserved."
@Article{DHKM2008,
AUTHOR = {Dress, Andreas W. M. and Huber, Katharina and Koolen, Jacobus and Moulton, Vincent},
TITLE = {Compatible decompositions and block realizations of finite metrics},
YEAR = {2008},
JOURNAL = {EJC},
VOLUME = {29},
NUMBER = {7},
PAGES = {1617-1633},
URL = {http://dx.doi.org/10.1016/j.ejc.2007.10.003},
NOTE = { http://www.ims.nus.edu.sg/preprints/2007-21.pdf},
ANNOTE = {BIBUPDATE : 20080303},
KEYWORDS = {abstract network, block realization, from distances, phylogenetic network, phylogeny, realization, reconstruction} } |
- forked on GitHub.