Thin Quartet Sets

Electronics - Devices

Oct 7, 2013 (3 years and 2 months ago)

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Quaret Compatibility:

New Results and Surprising Counterexamples

Stefan Gr
ünewald

Compatibility Problem

Given

a

set

of

phylogenetic

trees

with

overlapping

taxa

sets,

does

there

exist

a

single

phylogenetic

tree

on

the

union

of

the

taxa

sets

that

contains

the

information

of

all

input

trees?

Restrictions

A
restriction
of a phylogenetic tree
T

to a
subset
S

of
L
(
T
)

is the tree obtained from the
smallest subtree of
T

containing
S

by
suppressing all vertices of degree 2.

b

g

h

c

e

f

a

d

Restrictions

A
restriction
of a phylogenetic tree
T

to a
subset
S
of
L
(
T
)

is the tree obtained from the
smallest subtree of
T

containing
S

by
suppressing all vertices of degree 2.

g

h

c

e

d

Restrictions

A
restriction
of a phylogenetic tree
T

to a
subset
S

of
L
(
T
)

is the tree obtained from the
smallest subtree of
T

containing
S

by
suppressing all vertices of degree 2.

g

h

c

e

d

Displaying

T

displays
a binary tree

T

if

and
T

is the restriction of
T

to
L
(
T
’).

b

g

h

c

e

f

a

d

Displaying

T

displays
a binary tree

T

if

and
T

is the restriction of
T

to
L
(
T
’).

b

g

h

c

e

f

a

d

Displayed

c

d

a

h

Displaying

T

displays
a binary tree

T

if

and
T

is the restriction of
T

to
L
(
T
’).

b

g

h

c

e

f

a

d

Not displayed

a

f

b

h

Quartets

A
quartet
is a binary phylogenetic tree with
4 taxa.

The quartet separating
a
and
b
from
c
and
d
is denoted by
ab
|
cd.

ab
|
cd

ac
|
bd

|
bc

a

d

b

c

a

c

b

d

a

b

c

d

Compatibility

For a set
P

of binary phylogenetic trees, let
L
(
P
)

be the union of the taxa sets of all trees
in
P
.

P

is
compatible

if there is a tree
T

with
L
(
T
)
=L
(
P
)

such that
T

displays every tree in
P
.

If
T

is unique then
P

defines

T
.

Compatibility

It is NP
-
complete to decide if a given set of
binary phylogenetic trees is compatible, even if
all trees are quartets (Steel 1992).

It can be decided in polynomial time if a set
Q
of
quartets with
|Q|=|L
(
Q
)
|
-
3

defines a phylogenetic
tree (B
öcker, Dress, Steel 1999)
.

Given a set
Q
of quartets and a tree
T

that
displays
Q.
The complexity of deciding if
Q
defines
T

is unknown.

Thin Quartet Sets

A set
Q

of quartets is
thin
if every subset of
the taxa set with
k
≥4 elements contains at
most
k
-
3 quartets.

Theorem 1:
If
Q

is thin then
Q

is compatible.

Maximal Hierarchies

Theorem 2 (B
öcker, Dress, Steel, 1999):

Every
minimum defining quartet set contains
a maximal hierarchy of excess
-
free subsets.

Dezulian, Steel (2004):

“… one of the most mysterious and
apparently difficult results in phylogeny.”

Maximal Hierarchies

bg|af

ah|bf

cd|gh

cf|dg

Maximal Hierarchies

f

g

h

b

a

bg|af

ah|bf

cd|gh

cf|dg

Maximal Hierarchies

f

g

h

b

c

f

g

h

d

a

bg|af

ah|bf

cd|gh

cf|dg

Maximal Hierarchies

c

f

g

h

b

a

f

g

h

b

c

f

g

h

d

a

d

bg|af

ah|bf

cd|gh

cf|dg

freely compatible
if, for every
choice of one quartet for each quadruple, the
obtained quartet set is compatible.

thin
if every subset of the taxa
set with
k
≥4 elements contains at most
k
-
3

By Theorem 1, every thin set of quadruples is freely
compatible.

Question:
Is every freely compatible set of

By Theorem 1, every thin set of quadruples is freely
compatible.

Question:
Is every freely compatible set of

A set of 13 quadruples on 13 taxa such that every
two quadruples intersect in exactly two taxa is a
counterexample (SG, Li Junrui).

The Quartet Graph

a

b

c

d

e

f

ab|de, bc|ef, cd|fa

The Quartet Graph

a

b

c

d

e

f

ab|de, bc|ef, cd|fa

a

b

c

d

e

f

The Quartet Graph

a,b

c

d

e

f

ab|de, bc|ef, cd|fa

a

b

c

d

e

f

The Quartet Graph

a,b

c

d

e,f

ab|de, bc|ef, cd|fa

a

b

c

d

e

f

The Quartet Graph

a,b

c,d

e,f

ab|de, bc|ef, cd|fa

a

b

c

d

e

f

Closure Rules

a

b

c

d

e

ab|cd, bc|de
infer
ab|ce, ab|de, ac|de.

A quartet set is called

closed
if no closure rule can be
applied to any subset.

An Example

The quartet set corresponding to the picture is
strongly closed but not compatible (SG, Steel,
Swenson, 2007).

An incompatible quartet set
with little overlap

In1992, Mike Steel used a counting argument to show
that are incompatible quartet sets such that two quartets
in the set have at most one taxon in common.

An Example

An Example