1.
WG'07 - Dornburg
On restrictions of
balanced 2-interval graphs
Philippe Gambette and Stéphane Vialette
2.
Outline
• Introduction on 2-interval graphs
• Motivations for the study of this class
• Balanced 2-interval graphs
• Unit 2-interval graphs
• Investigating unit 2-interval graph recognition
3.
2-interval graphs
2-interval graphs are intersection graphs of pairs of intervals
a vertex a pair of intervals
8
1 2
5
I 3
4
6 9
7
the pairs of intervals
an edge
have a non-empty
between two
vertices intersection
5 8
1
9
G 4
2
3
6 7
I is a realization of 2-interval graph G.
4.
Why consider 2-interval graphs?
A 2-interval can represent :
- a task split in two parts in scheduling
When two tasks are scheduled in the same time,
corresponding nodes are adjacent.
5.
Why consider 2-interval graphs?
A 2-interval can represent :
- a task split in two parts in scheduling
- similar portions of DNA in DNA comparison
The aim is to find a large set of non overlapping similar
portions, that is a large independent set in the 2-interval graph.
6.
Why consider 2-interval graphs?
A 2-interval can represent:
- a task split in two parts in scheduling
- similar portions of DNA in DNA comparison
- complementary portions of RNA in RNA secondary
structure prediction
Primary structure:
AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU
Secondary structure: CU
C A
CG
GC 2
A
G
GAU
U U U
C C
AGUA U
C
A U
1 C U G
G C C C AC
U
UC
3
7.
RNA secondary structure prediction
U A
Helices: sets of contiguous base
U A
pairs, appearing successive, or
C C
A nested, in the primary structure.
U
A U
C I2 I3 I1
G U
C
I2 C I2
G
A successive nested
U
C
U G
UUCGU Find the maximum set of disjoint
C
G successive or nested 2-intervals:
G
AAGCA dynamic programming.
U
C UC CG
C
I1 A A
G
I 3
A
helices C
GU
G
U
G G
U
A
8.
RNA secondary structure prediction
Pseudo-knot: crossing I1
base pairs.
I1 I2
crossed I2
5' extremity or the RNA
component of human telomerase
From D.W. Staple, S.E. Butcher,
Pseudoknots: RNA structures with Diverse Functions
(PloS Biology 2005 3:6 p.957)
9.
Why consider 2-interval graphs?
A 2-interval can represent:
- a task split in two parts in scheduling
- similar portions of DNA in DNA comparison
- complementary portions of RNA in RNA secondary
structure prediction AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU
8
1 2
5
3
4
6 9
CU 7
C A
CG
GC 2 5
A 8
1
G
GAU
U U U
C 9
C
AGUA U
C
A 4
U
1 C 2
U G
G C C C AC 3
U
UC
3 6 7
10.
Why consider 2-interval graphs?
A 2-interval can represent: Both intervals
have same size!
- a task split in two parts in scheduling
- similar portions of DNA in DNA comparison
- complementary portions of RNA in RNA secondary
structure prediction AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU
8
1 2
5
3
4
6 9
CU 7
C A
CG
GC 2 5
A 8
1
G
GAU
U U U
C 9
C
AGUA U
C
A 4
U
1 C 2
U G
G C C C AC 3
U
UC
3 6 7
11.
Restrictions of 2-interval graphs
We introduce restrictions on 2-intervals:
- both intervals of a 2-interval have same size:
balanced 2-interval graphs
- all intervals have the same length:
unit 2-interval graphs
- all intervals are open, have integer coordinates, and length x:
(x,x)-interval graphs
12.
Inclusion of graph classes
perfect
2-inter AT-free
K1,4-free
circle
co-compar compar
Ko
sto
ch
ka
claw-free ,W
es chordal
t, 1
99
9
trapezoid
circ-arc
outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
Following ISGCI
13.
Some properties of 2-interval graphs
Recognition: NP-hard (West and Shmoys, 1984)
Coloring: NP-hard from line graphs
Maximum Independent Set: NP-hard
(Bafna et al, 1996; Vialette, 2001)
Maximum Clique: open, NP-complete on 3-interval graphs
(Butman et al, 2007)
14.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
chordal
trapezoid
circ-arc
outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
15.
Balanced 2-interval graphs
2-interval graphs do not all have a balanced realization.
Proof:
Idea: a cycle of three 2-intervals which induce a contradiction.
I1
I2 I3
B3 B4
B1 B2
B5 B6
l (I 2) < l (I 1) l (I 3) < l (I 2)
l (I 3) < l (I 1) l (I 1) < l (I 3)
Build a graph where something of length>0 (a hole between
two intervals) is present inside each box Bi.
16.
Balanced 2-interval graphs
2-interval graphs do not all have a balanced realization.
Proof:
Gadget: K5,3, every 2-interval realization of K5,3 is a
contiguous set of intervals (West and Shmoys, 1984)
has only « chained » realizations:
17.
Balanced 2-interval graphs
2-interval graphs do not all have a balanced realization.
Proof:
Gadget: K5,3, every 2-interval realization of K5,3 is a
contiguous set of intervals (West and Shmoys, 1984)
has only « chained » realizations:
18.
Balanced 2-interval graphs
2-interval graphs do not all have a balanced realization.
Proof:
Example of 2-interval graph with no balanced realization:
has only unbalanced realizations:
I1 I2 I3
19.
Recognition of balanced 2-interval graphs
Recognizing balanced 2-interval
graphs is NP-complete.
Idea of the proof:
Adapt the proof by West and Shmoys using balanced gadgets.
A balanced realization of K5,3:
length: 79
20.
Recognition of balanced 2-interval graphs
Recognizing balanced 2-interval
graphs is NP-complete.
Idea of the proof:
Reduction of Hamiltonian Cycle on triangle-free 3-regular
graphs, which is NP-complete (West, Shmoys, 1984).
21.
Recognition of balanced 2-interval graphs
For any 3-regular triangle-free graph G, build in polynomial
time a graph G' which has a 2-interval realization (which is
balanced) iff G has a Hamiltonian cycle.
Idea: if G has a Hamiltonian cycle, add gadgets on G to get G'
and force that any 2-interval realization of G' can be split into
intervals for the Hamiltonian cycle and intervals for a perfect
matching.
= U
G
depth 2
22.
Recognition of balanced 2-interval graphs
Recognizing balanced 2-interval
graphs is NP-complete.
G'
v0 v1
z
M(v1)
M(v0)
H1 H2
H3
23.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
chordal
trapezoid
circ-arc
outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
24.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
chordal
trapezoid
circ-arc
outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
25.
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
26.
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
27.
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
28.
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
29.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
chordal
trapezoid
circ-arc
outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
30.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
unit-2-inter
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
31.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
32.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
Take the left-most and the one it intersects.
33.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
Increment their length to the right and translate the ones on
the right.
34.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
Take the left-most and the one it intersects.
35.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
Increment their length to the right and translate the ones on
the right.
36.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
37.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
38.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
39.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
40.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
41.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of inclusion:
How to transform a (x,x)-realization into a
(x+1,x+1)-realization?
Consider each interval separately.
42.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
Proof of strictness:
Gadget: K4,4-e, every 2-interval realization of K4,4-e is a
contiguous set of intervals.
I5
I1
I8 I5 I6 I7
I6
I2
I7
I3
I1 I2 3 4
II
I4 I8
K4,4-e has a (2,2)-interval realization!
43.
(x,x)-interval graphs
The class of (x,x)-interval graphs is strictly included
in the class of (x+1,x+1)-interval graphs for x>1.
G4
a v1 v'1
Idea of the proof of strictness:
X1 X2
For x=4: any 2-interval vl1 vr2
realization of G4 has two
1 2
v2 v'2
v v
r l
vl4 vr4 3
vr3
v
“stairways” which requires l
v3 v'3
X4 X3
“steps” of length at least 5.
v4 v'4
b
v1 v'1
v2 v'2
v3 v'3
v4
vl1 vr4
X4
v'4
X1
vl3
vr1 vr3 vl4
X3
vl2 vr2
X2
a b
44.
(x,x)-interval graphs
{unit 2-interval graphs} = U {(x,x)-interval graphs}
x>0
Proof of the inclusion:
There is a linear algorithm to compute a realization of a
unit interval graph where interval endpoints are rational,
with denominator 2n (Corneil et al, 1995).
Corollary:
If recognizing (x,x)-interval graphs is polynomial for all x
then recognizing unit 2-interval graphs is polynomial.
45.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
unit-2-inter
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
46.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
unit-2-inter
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
47.
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
48.
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
49.
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
50.
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
proper = unit
51.
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
+ disjoint intervals
52.
Inclusion of graph classes
perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
co-compar compar
claw-free
unit-2-inter
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
53.
Inclusion of graph classes
Quasi-line graphs: every vertex is AT-free perfect
2-inter
bisimplicial (its neighborhood can
be partitioned into 2 cliques).
balanced
2-inter
K1,4-free
circle
co-compar compar
unit-2-inter
claw-free
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
quasi-line
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
54.
Inclusion of graph classes
Quasi-line graphs: every vertex is AT-free perfect
2-inter
bisimplicial (its neighborhood can
be partitioned into 2 cliques).
balanced
2-inter
K1,4-free
circle
co-compar compar
unit-2-inter
claw-free
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
quasi-line
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
55.
Inclusion of graph classes
K1,5-free perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
all-4-simp
co-compar compar
unit-2-inter
claw-free
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
quasi-line
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
56.
Recognition of all-k-simplicial graphs
A graph is all-k-simplicial if the neighborhood of a vertex can
be partitioned in at most k cliques.
Recognizing all-k-simplicial graphs is NP-complete for k>2.
Proof:
Reduction from k-colorability.
G k-colorable iff G' all-k-simplicial, where G' is the
complement graph of G
+ 1 universal vertex
G G'
57.
Inclusion of graph classes
K1,5-free perfect
2-inter AT-free
balanced
2-inter
K1,4-free
circle
all-4-simp
co-compar compar
unit-2-inter
claw-free
chordal
trapezoid
circ-arc
(2,2)-inter outerplanar
odd-anti
quasi-line
cycle-free
bipartite
proper circ-arc
= circ. interval
line
unit interval
circ-arc
co-comp int.
unit = proper trees
permutation
middle dim 2 height 1
interval
58.
Unit 2-interval graph recognition
Complexity still open.
Algorithm and characterization for bipartite graphs:
A bipartite graph is a unit 2-interval graph
(and a (2,2)-interval graph) iff it has
maximum degree 4 and is not 4-regular.
Linear algorithm based on finding paths in the graph and
orienting and joining them.
59.
Perspectives
Recognition of unit 2-interval graphs and
(x,x)-interval graphs remains open.
The maximum clique problem is still open on
2-interval graphs and restrictions.
60.
Perspectives
Recognition of unit 2-interval graphs and
(x,x)-interval graphs remains open.
The maximum clique problem is still open on
2-interval graphs and restrictions.
Guten Appetit!
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