157,105 research outputs found
Growth of the Brownian forest
Trees in Brownian excursions have been studied since the late 1980s. Forests
in excursions of Brownian motion above its past minimum are a natural extension
of this notion. In this paper we study a forest-valued Markov process which
describes the growth of the Brownian forest. The key result is a composition
rule for binary Galton--Watson forests with i.i.d. exponential branch lengths.
We give elementary proofs of this composition rule and explain how it is
intimately linked with Williams' decomposition for Brownian motion with drift.Comment: Published at http://dx.doi.org/10.1214/009117905000000422 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Complexity of Splits Reconstruction for Low-Degree Trees
Given a vertex-weighted tree T, the split of an edge xy in T is min{s_x(xy),
s_y(xy)} where s_u(uv) is the sum of all weights of vertices that are closer to
u than to v in T. Given a set of weighted vertices V and a multiset of splits
S, we consider the problem of constructing a tree on V whose splits correspond
to S. The problem is known to be NP-complete, even when all vertices have unit
weight and the maximum vertex degree of T is required to be no more than 4. We
show that the problem is strongly NP-complete when T is required to be a path,
the problem is NP-complete when all vertices have unit weight and the maximum
degree of T is required to be no more than 3, and it remains NP-complete when
all vertices have unit weight and T is required to be a caterpillar with
unbounded hair length and maximum degree at most 3. We also design polynomial
time algorithms for the variant where T is required to be a path and the number
of distinct vertex weights is constant, and the variant where all vertices have
unit weight and T has a constant number of leaves. The latter algorithm is not
only polynomial when the number of leaves, k, is a constant, but also
fixed-parameter tractable when parameterized by k. Finally, we shortly discuss
the problem when the vertex weights are not given but can be freely chosen by
an algorithm.
The considered problem is related to building libraries of chemical compounds
used for drug design and discovery. In these inverse problems, the goal is to
generate chemical compounds having desired structural properties, as there is a
strong correlation between structural properties, such as the Wiener index,
which is closely connected to the considered problem, and biological activity
The agreement distance of unrooted phylogenetic networks
A rearrangement operation makes a small graph-theoretical change to a
phylogenetic network to transform it into another one. For unrooted
phylogenetic trees and networks, popular rearrangement operations are tree
bisection and reconnection (TBR) and prune and regraft (PR) (called subtree
prune and regraft (SPR) on trees). Each of these operations induces a metric on
the sets of phylogenetic trees and networks. The TBR-distance between two
unrooted phylogenetic trees and can be characterised by a maximum
agreement forest, that is, a forest with a minimum number of components that
covers both and in a certain way. This characterisation has
facilitated the development of fixed-parameter tractable algorithms and
approximation algorithms. Here, we introduce maximum agreement graphs as a
generalisations of maximum agreement forests for phylogenetic networks. While
the agreement distance -- the metric induced by maximum agreement graphs --
does not characterise the TBR-distance of two networks, we show that it still
provides constant-factor bounds on the TBR-distance. We find similar results
for PR in terms of maximum endpoint agreement graphs.Comment: 23 pages, 13 figures, final journal versio
On the strictness of the quantifier structure hierarchy in first-order logic
We study a natural hierarchy in first-order logic, namely the quantifier
structure hierarchy, which gives a systematic classification of first-order
formulas based on structural quantifier resource. We define a variant of
Ehrenfeucht-Fraisse games that characterizes quantifier classes and use it to
prove that this hierarchy is strict over finite structures, using strategy
compositions. Moreover, we prove that this hierarchy is strict even over
ordered finite structures, which is interesting in the context of descriptive
complexity.Comment: 38 pages, 8 figure
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
Fractional colorings of cubic graphs with large girth
We show that every (sub)cubic n-vertex graph with sufficiently large girth
has fractional chromatic number at most 2.2978 which implies that it contains
an independent set of size at least 0.4352n. Our bound on the independence
number is valid to random cubic graphs as well as it improves existing lower
bounds on the maximum cut in cubic graphs with large girth
- …