573 research outputs found
Composing dynamic programming tree-decomposition-based algorithms
Given two integers and as well as graph classes
, the problems
,
, and
ask, given graph
as input, whether , , respectively can be partitioned
into sets such that, for each between and
, , , respectively. Moreover in , we request that the number of edges with
endpoints in different sets of the partition is bounded by . We show that if
there exist dynamic programming tree-decomposition-based algorithms for
recognizing the graph classes , for each , then we can
constructively create a dynamic programming tree-decomposition-based algorithms
for ,
, and
. We show that, in
some known cases, the obtained running times are comparable to those of the
best know algorithms
Leaf-reconstructibility of phylogenetic networks
An important problem in evolutionary biology is to reconstruct the
evolutionary history of a set of species. This history is often represented
as a phylogenetic network, that is, a connected graph with leaves labelled by
elements in (for example, an evolutionary tree), which is usually also
binary, i.e. all vertices have degree 1 or 3. A common approach used in
phylogenetics to build a phylogenetic network on involves constructing it
from networks on subsets of . Here we consider the question of which
(unrooted) phylogenetic networks are leaf-reconstructible, i.e. which networks
can be uniquely reconstructed from the set of networks obtained from it by
deleting a single leaf (its -deck). This problem is closely related to the
(in)famous reconstruction conjecture in graph theory but, as we shall show,
presents distinct challenges. We show that some large classes of phylogenetic
networks are reconstructible from their -deck. This includes phylogenetic
trees, binary networks containing at least one non-trivial cut-edge, and binary
level-4 networks (the level of a network measures how far it is from being a
tree). We also show that for fixed , almost all binary level-
phylogenetic networks are leaf-reconstructible. As an application of our
results, we show that a level-3 network can be reconstructed from its
quarnets, that is, 4-leaved networks that are induced by in a certain
recursive fashion. Our results lead to several interesting open problems which
we discuss, including the conjecture that all phylogenetic networks with at
least five leaves are leaf-reconstructible
Treewidth, crushing, and hyperbolic volume
We prove that there exists a universal constant such that any closed
hyperbolic 3-manifold admits a triangulation of treewidth at most times its
volume. The converse is not true: we show there exists a sequence of hyperbolic
3-manifolds of bounded treewidth but volume approaching infinity. Along the
way, we prove that crushing a normal surface in a triangulation does not
increase the carving-width, and hence crushing any number of normal surfaces in
a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former
argument (in V1) used a construction that relied on a wrong theorem. Section
5.1 has also been adjusted to the new construction. Various other arguments
have been clarifie
Compatibility of unrooted phylogenetic trees is FPT
Abstract A collection of T 1 , T 2 , . . . , T k of unrooted, leaf labelled (phylogenetic) trees, all with different leaf sets, is said to be compatible if there exists a tree T such that each tree T i can be obtained from T by deleting leaves and contracting edges. Determining compatibility is NP-hard, and the fastest algorithm to date has worst case complexity of around (n k ) time, n being the number of leaves. Here, we present an O(nf (k)) algorithm, proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable (FPT) with respect to the number k of trees
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