On restrictions of balanced 2-interval graphs

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.


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.


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)



- 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



- 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)


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


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.


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:



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:



has only unbalanced realizations:I 1 I 2 I 3

Proof:

Example of 2-interval graph with no 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


Idea of the proof:

Reduction of Hamiltonian Cycle on triangle-free 3-regular graphs, which is NP-complete (West, Shmoys, 1984).



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



zM(v

1)M(v

0)

H1

H2

H3

G'

v1v

0


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


balanced2-inter


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


balanced2-inter

Circular-arc and balanced 2-interval graphs

Circular-arc graphs are balanced 2-interval graphs

Proof:



Proof:



Proof:



Proof:


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


balanced2-inter


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


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.



Proof of inclusion:



Take the left-most and the one it intersects.



Proof of inclusion:



Increment their length to the right and translate the ones on the right.



Proof of inclusion:



Take the left-most and the one it intersects.



Proof of inclusion:



Increment their length to the right and translate the ones on the right.



Proof of inclusion:





Proof of inclusion:





Proof of inclusion:





Proof of inclusion:





Proof of inclusion:





Proof of inclusion:





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!



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


{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:


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


balanced2-inter

(2,2)-inter

unit-2-inter


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


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:



Proof:



Proof:

proper = unit



Proof:

+ disjoint intervals



Proof:


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


balanced2-inter

(2,2)-inter

unit-2-inter


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


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).


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free


unitcirc-arc


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).


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free

K1,5

-free

(2,2)-inter

unit-2-inter

balanced2-inter

quasi-line


all-4-simp

unitcirc-arc


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.


chordal

trees

compar

permutation

co-compar

trapezoid

bipartite

2-inter AT-free

lineinterval

circ-arc

circle

outerplanar


claw-free

odd-anticycle-free

K1,4

-free

K1,5

-free

(2,2)-inter

unit-2-inter

balanced2-inter

quasi-line


all-4-simp

unitcirc-arc


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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On restrictions of balanced 2-interval graphs