on restrictions of balanced 2-interval graphs
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WG'07 - Dornburg
On restrictions ofbalanced 2-interval graphs
Philippe Gambette and Stéphane Vialette
• Balanced 2-interval graphs• Unit 2-interval graphs
Outline
• Introduction on 2-interval graphs• Motivations for the study of this class
• Investigating unit 2-interval graph recognition
2-interval graphs
I is a realization of 2-interval graph G.
a vertex a pair of intervals
an edgebetween two
vertices
the pairs of intervals have a non-empty intersection
2-interval graphs are intersection graphs of pairs of intervals
I1
5
64
79
2 8
3
7
49
1 5 8
32
6
G
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.
Why consider 2-interval graphs?
A 2-interval can represent :
- a task split in two parts in scheduling- similar portions of DNA in DNA comparisonThe aim is to find a large set of non overlapping similar portions, that is a large independent set in the 2-interval graph.
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 predictionPrimary structure:
Secondary structure:
A G G U AGC
CC U
AGCU
C
U C C A
G C C
U
U
A
C
G
A
U
C
A
U CU
UUC
G
AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU
1
2
3
RNA secondary structure prediction
AA
CG
CUA
U U C G U
A A G C A
CU
U AAC
UUCUC
GUG
CG
CC U CAG
GUC G
AAC
I 1
I 3
I 2
helices
GGG
U
UUG
Helices: sets of contiguous base pairs, appearing successive, or nested, in the primary structure.
I 2 I 3 I 1
I 2
successive nested
Find the maximum set of disjoint successive or nested 2-intervals: dynamic programming.
A
RNA secondary structure prediction
Pseudo-knot: crossing base pairs.
I 1 I 2
crossed
I 1
I 2
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)
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
7
49
1A G G U AGC
CC U
AGCU
C
U C C A
G C C
U
U
A
C
G
A
U
C
A
U CU
UUC
G
AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU1
5
64
79
2 8
3
5 8
32
6
1
2
3
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
7
49
1A G G U AGC
CC U
AGCU
C
U C C A
G C C
U
U
A
C
G
A
U
C
A
U CU
UUC
G
AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU1
5
64
79
2 8
3
5 8
32
6
1
2
3
Both intervals have same size!
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
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
Kostochka, West, 1999
Following ISGCI
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)
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
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.
I 1
I 2
B1
B2
B3
B4
B5
B6
I 3
l (I 2) < l (I
1) l (I 3) < l (I
2)
l (I 1) < l (I
3)l (I 3) < l (I
1)
Build a graph where something of length>0 (a hole between two intervals) is present inside each box B
i.
Balanced 2-interval graphs
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:
2-interval graphs do not all have a balanced realization.
Balanced 2-interval graphs
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:
2-interval graphs do not all have a balanced realization.
Balanced 2-interval graphs
has only unbalanced realizations:I 1 I 2 I 3
Proof:
Example of 2-interval graph with no balanced realization:
2-interval graphs do not all have a balanced realization.
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
Recognition of balanced 2-interval graphs
Recognition of balanced 2-interval graphs
Idea of the proof:
Reduction of Hamiltonian Cycle on triangle-free 3-regular graphs, which is NP-complete (West, Shmoys, 1984).
Recognizing balanced 2-interval graphs is NP-complete.
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.
G U=
depth 2
Recognition of balanced 2-interval graphs
Recognizing balanced 2-interval graphs is NP-complete.
zM(v
1)M(v
0)
H1
H2
H3
G'
v1v
0
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
Circular-arc and balanced 2-interval graphs
Circular-arc graphs are balanced 2-interval graphs
Proof:
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
(2,2)-inter
unit-2-inter
(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.
(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.
(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.
(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.
(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.
(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.
(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.
(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.
(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.
(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.
(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.
(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.
I 1 I 2I 3 I 4
I 8I 5 I
6 I
7
I 1
I 6
I 7
I 8
I 5
I 2
I 3
I 4
K4,4
-e has a (2,2)-interval realization!
(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.
Idea of the proof of strictness:
For x=4: any 2-interval realization of G
4 has two
“stairways” which requires “steps” of length at least 5.
v4
v'4
X4
X3
X1 X
2
v3
v'3
v2
v'2
v1
v'1
vl1
vr4v
l4 v
r3
vr1 v
l2 v
r2
vl3
b
a
X3
X4
X2
vl2
vr3v
r1
v1
vr2
v2v
3v4
v'1v'
2
v'4
v'3
vl3
vl4
X1
vl1 v
r4
a b
G4
(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).
If recognizing (x,x)-interval graphs is polynomial for all x then recognizing unit 2-interval graphs is polynomial.
Corollary:
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
(2,2)-inter
unit-2-inter
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
(2,2)-inter
unit-2-inter
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
proper = unit
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
+ disjoint intervals
Proper circular-arc and unit 2-interval graphs
Proper circular-arc graphs are unit 2-interval graphs
Proof:
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
(2,2)-inter
unit-2-inter
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
(2,2)-inter
unit-2-inter
quasi-line
Quasi-line graphs: every vertex is bisimplicial (its neighborhood can
be partitioned into 2 cliques).
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
proper circ-arc= circ. interval
unitcirc-arc
unit = properintervalmiddle
balanced2-inter
(2,2)-inter
unit-2-inter
quasi-line
Quasi-line graphs: every vertex is bisimplicial (its neighborhood can
be partitioned into 2 cliques).
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
K1,5
-free
(2,2)-inter
unit-2-inter
balanced2-inter
quasi-line
proper circ-arc= circ. interval
all-4-simp
unitcirc-arc
unit = properintervalmiddle
Recognition of all-k-simplicial graphs
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 thecomplement graph of G+ 1 universal vertex
G G'
A graph is all-k-simplicial if the neighborhood of a vertex can be partitioned in at most k cliques.
Inclusion of graph classesperfect
chordal
trees
compar
permutation
co-compar
trapezoid
bipartite
2-inter AT-free
lineinterval
circ-arc
circle
outerplanar
co-comp int.dim 2 height 1
claw-free
odd-anticycle-free
K1,4
-free
K1,5
-free
(2,2)-inter
unit-2-inter
balanced2-inter
quasi-line
proper circ-arc= circ. interval
all-4-simp
unitcirc-arc
unit = properintervalmiddle
Unit 2-interval graph recognition
Complexity still open.
Algorithm and characterization for bipartite graphs:
Linear algorithm based on finding paths in the graph and orienting and joining them.
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.
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.
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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