WG'07 - Dornburg


      On restrictions of
  balanced 2-interval graphs
Philippe Gambette and Stéphane Vialette
Outline

• Introduction on 2-interval graphs
• Motivations for the study of this class
• Balanced 2-interval graphs
• Unit...
2-interval graphs
2-interval graphs are intersection graphs of pairs of intervals

                a vertex               ...
Why consider 2-interval graphs?
A 2-interval can represent :
- a task split in two parts in scheduling

When two tasks are...
Why consider 2-interval graphs?
A 2-interval can represent :
- a task split in two parts in scheduling
- similar portions ...
Why consider 2-interval graphs?
A 2-interval can represent:
- a task split in two parts in scheduling
- similar portions o...
RNA secondary structure prediction
                U   A
                              Helices: sets of contiguous base
  ...
RNA secondary structure prediction



 Pseudo-knot: crossing                    I1
 base pairs.

     I1             I2

 ...
Why consider 2-interval graphs?
A 2-interval can represent:
- a task split in two parts in scheduling
- similar portions o...
Why consider 2-interval graphs?
A 2-interval can represent:                                           Both intervals
     ...
Restrictions of 2-interval graphs
We introduce restrictions on 2-intervals:
- both intervals of a 2-interval have same siz...
Inclusion of graph classes
                                                                                     perfect
  ...
Some properties of 2-interval graphs

Recognition: NP-hard (West and Shmoys, 1984)
Coloring: NP-hard from line graphs
Maxi...
Inclusion of graph classes
                                                                              perfect
    2-int...
Balanced 2-interval graphs
 2-interval graphs do not all have a balanced realization.
Proof:
Idea: a cycle of three 2-inte...
Balanced 2-interval graphs
 2-interval graphs do not all have a balanced realization.
Proof:
Gadget: K5,3, every 2-interva...
Balanced 2-interval graphs
 2-interval graphs do not all have a balanced realization.
Proof:
Gadget: K5,3, every 2-interva...
Balanced 2-interval graphs
 2-interval graphs do not all have a balanced realization.
Proof:
Example of 2-interval graph w...
Recognition of balanced 2-interval graphs

             Recognizing balanced 2-interval
                graphs is NP-compl...
Recognition of balanced 2-interval graphs

            Recognizing balanced 2-interval
               graphs is NP-complet...
Recognition of balanced 2-interval graphs
For any 3-regular triangle-free graph G, build in polynomial
time a graph G' whi...
Recognition of balanced 2-interval graphs

        Recognizing balanced 2-interval
           graphs is NP-complete.
     ...
Inclusion of graph classes
                                                                              perfect
    2-int...
Inclusion of graph classes
                                                                              perfect
    2-int...
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
    2-int...
Inclusion of graph classes
                                                                                perfect
      2...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

   The class of (x,x)-interval graphs is strictly included
     in the class of (x+1,x+1)-interval ...
(x,x)-interval graphs

      The class of (x,x)-interval graphs is strictly included
        in the class of (x+1,x+1)-int...
(x,x)-interval graphs

   {unit 2-interval graphs} = U {(x,x)-interval graphs}
                               x>0


 Proof...
Inclusion of graph classes
                                                                                perfect
      2...
Inclusion of graph classes
                                                                                perfect
      2...
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 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:




     ...
Inclusion of graph classes
                                                                                perfect
      2...
Inclusion of graph classes
                 Quasi-line graphs: every vertex is AT-free                                 per...
Inclusion of graph classes
                 Quasi-line graphs: every vertex is AT-free                                 per...
Inclusion of graph classes
                                  K1,5-free                                                 per...
Recognition of all-k-simplicial graphs
A graph is all-k-simplicial if the neighborhood of a vertex can
be partitioned in a...
Inclusion of graph classes
                                  K1,5-free                                                 per...
Unit 2-interval graph recognition

Complexity still open.
Algorithm and characterization for bipartite graphs:

        A ...
Perspectives


       Recognition of unit 2-interval graphs and
         (x,x)-interval graphs remains open.


     The ma...
Perspectives


       Recognition of unit 2-interval graphs and
         (x,x)-interval graphs remains open.


     The ma...
Upcoming SlideShare
Loading in...5
×

On restrictions of balanced 2-interval graphs

1,183

Published on

A presentation on some subclasses of 2-interval graphs (WG 2007, Dornburg)

Published in: Education, Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
1,183
On Slideshare
0
From Embeds
0
Number of Embeds
3
Actions
Shares
0
Downloads
5
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

On restrictions of balanced 2-interval graphs

  1. 1. WG'07 - Dornburg On restrictions of balanced 2-interval graphs Philippe Gambette and Stéphane Vialette
  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 25. Circular-arc and balanced 2-interval graphs Circular-arc graphs are balanced 2-interval graphs Proof:
  26. 26. Circular-arc and balanced 2-interval graphs Circular-arc graphs are balanced 2-interval graphs Proof:
  27. 27. Circular-arc and balanced 2-interval graphs Circular-arc graphs are balanced 2-interval graphs Proof:
  28. 28. Circular-arc and balanced 2-interval graphs Circular-arc graphs are balanced 2-interval graphs Proof:
  29. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 47. Proper circular-arc and unit 2-interval graphs Proper circular-arc graphs are unit 2-interval graphs Proof:
  48. 48. Proper circular-arc and unit 2-interval graphs Proper circular-arc graphs are unit 2-interval graphs Proof:
  49. 49. Proper circular-arc and unit 2-interval graphs Proper circular-arc graphs are unit 2-interval graphs Proof:
  50. 50. Proper circular-arc and unit 2-interval graphs Proper circular-arc graphs are unit 2-interval graphs Proof: proper = unit
  51. 51. Proper circular-arc and unit 2-interval graphs Proper circular-arc graphs are unit 2-interval graphs Proof: + disjoint intervals
  52. 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. 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. 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. 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. 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. 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. 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. 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. 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!
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×