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
ad

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
bgaf
ahbf
cdgh
cfdg
Maximal Hierarchies
f
g
h
b
a
bgaf
ahbf
cdgh
cfdg
Maximal Hierarchies
f
g
h
b
c
f
g
h
d
a
bgaf
ahbf
cdgh
cfdg
Maximal Hierarchies
c
f
g
h
b
a
f
g
h
b
c
f
g
h
d
a
d
bgaf
ahbf
cdgh
cfdg
Freely compatible quadruple sets
A set of quadruples is
freely compatible
if, for every
choice of one quartet for each quadruple, the
obtained quartet set is compatible.
A set of quadruples is
thin
if every subset of the taxa
set with
k
≥4 elements contains at most
k

3
quadruples.
Freely compatible quadruple sets
By Theorem 1, every thin set of quadruples is freely
compatible.
Question:
Is every freely compatible set of
quadruples thin?
Freely compatible quadruple sets
By Theorem 1, every thin set of quadruples is freely
compatible.
Question:
Is every freely compatible set of
quadruples thin?
Answer: No!
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
abde, bcef, cdfa
The Quartet Graph
a
b
c
d
e
f
abde, bcef, cdfa
a
b
c
d
e
f
The Quartet Graph
a,b
c
d
e
f
abde, bcef, cdfa
a
b
c
d
e
f
The Quartet Graph
a,b
c
d
e,f
abde, bcef, cdfa
a
b
c
d
e
f
The Quartet Graph
a,b
c,d
e,f
abde, bcef, cdfa
a
b
c
d
e
f
Closure Rules
a
b
c
d
e
abcd, bcde
infer
abce, abde, acde.
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
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