On restrictions of balanced 2-interval graphs Philippe Gambette and - - PowerPoint PPT Presentation

WG'07 - Dornburg

On restrictions of balanced 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 edge between two vertices the pairs of intervals have a non-empty intersection 2-interval graphs are intersection graphs of pairs of intervals I

1 5 6 4 7 9 2 8 3 7 4 9 1 5 8 3 2 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 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.

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: Secondary structure:

A G G U A G C C C U A G C U C U C C A G C C U U A C G A U C A U C U U U C G

AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU

1 2 3

RNA secondary structure prediction

A A C G C U A U U C G U A A G C A C U U A A C U U C U C G U G C G C C U CAG GUC G AAC I 1 I 3 I 2 helices G G G U U U G

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 4 9 1 A G G U A G C C C U A G C U C U C C A G C C U U A C G A U C A U C U U U C G AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU 1 5 6 4 7 9 2 8 3 5 8 3 2 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 4 9 1 A G G U A G C C C U A G C U C U C C A G C C U U A C G A U C A U C U U U C G AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU 1 5 6 4 7 9 2 8 3 5 8 3 2 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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle

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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-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 Bi.

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.

z M(v1) M(v0) H1 H2 H3 G' v1 v0

Inclusion of graph classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-inter

Inclusion of graph classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-inter

Inclusion of graph classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-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 2 I 3 I 4 I 8 I

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 G4 has two “stairways” which requires “steps” of length at least 5.

v4 v'4 X4 X3 X1 X2 v3 v'3 v2 v'2 v1 v'1 vl

1

vr

4

vl

4

vr

3

vr

1

vl

2

vr

2

vl

3

b a X3 X4 X2 vl

2

vr

3

vr

1

v1 vr

2

v2 v3 v4 v'1 v'2 v'4 v'3 vl

3

vl

4

X1 vl

1

vr

4

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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-inter (2,2)-inter unit-2-inter

Inclusion of graph classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-inter (2,2)-inter unit-2-inter

Inclusion of graph classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free proper circ-arc = circ. interval unit circ-arc unit = proper interval middle balanced 2-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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free K1,5-free (2,2)-inter unit-2-inter balanced 2-inter quasi-line proper circ-arc = circ. interval all-4-simp unit circ-arc unit = proper interval middle

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 the complement 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 classes

perfect chordal trees compar permutation co-compar trapezoid bipartite 2-inter AT-free line interval circ-arc circle

• uterplanar

co-comp int. dim 2 height 1 claw-free

• dd-anti

cycle-free K1,4-free K1,5-free (2,2)-inter unit-2-inter balanced 2-inter quasi-line proper circ-arc = circ. interval all-4-simp unit circ-arc unit = proper interval middle

Unit 2-interval graph recognition

Complexity still open. Algorithm and characterization for bipartite graphs: Linear algorithm based on finding paths in the graph and

• rienting 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!