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

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

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 between two have a non-empty vertices intersection 5 8 1 9 G 4 2 3 6 7 I is a realization of 2-interval graph 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: AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU Secondary structure: C U C A C G 2 G C A G G U A U U C C A U C U A A G U U 1 C U G C A G C C C U C U 3

RNA secondary structure prediction U A Helices : sets of contiguous base U A pairs, appearing successive, or C C A U nested, in the primary structure. A U C I 2 I 3 I 1 G U C I 2 I 2 C G A successive nested U C U G U U C G U Find the maximum set of disjoint C G successive or nested 2-intervals: GUC G A A G C A dynamic programming . C U C CAG I 1 AAC I 3 helices G U G U G G U A

RNA secondary structure prediction Pseudo-knot : crossing I 1 base pairs. I 1 I 2 crossed 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 AGGUAGCCCUAGCUUAGUACUUGUCUCACUCCGCACCU 8 1 2 5 3 4 6 9 C U 7 C A C G 2 5 A G G U A G C 8 1 U U C C A U 9 C U A A G U 4 U 1 C 2 U G C A G C C 3 C U C U 3 6 7

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

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 2-inter AT-free perfect K 1,4 -free circle co-compar compar Kostochka, West, 1999 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 permutation trees middle dim 2 height 1 interval 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 2-inter AT-free perfect balanced 2-inter K 1,4 -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 permutation trees middle dim 2 height 1 interval

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 I 3 B 3 B 4 B 1 B 2 B 5 B 6 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 B i .

Balanced 2-interval graphs 2-interval graphs do not all have a balanced realization. Proof: Gadget: K 5,3 , every 2-interval realization of K 5,3 is a contiguous set of intervals (West and Shmoys, 1984) has only « chained » realizations:

Balanced 2-interval graphs 2-interval graphs do not all have a balanced realization. Proof: Gadget: K 5,3 , every 2-interval realization of K 5,3 is a contiguous set of intervals (West and Shmoys, 1984) has only « chained » realizations:

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: I 1 I 2 I 3

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 K 5,3 : length: 79

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).

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

Recognition of balanced 2-interval graphs Recognizing balanced 2-interval graphs is NP-complete. G' v 1 v 0 z M ( v 1 ) M ( v 0 ) H 3 H 1 H 2

Inclusion of graph classes 2-inter AT-free perfect balanced 2-inter K 1,4 -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 permutation trees middle dim 2 height 1 interval

Inclusion of graph classes 2-inter AT-free perfect balanced 2-inter K 1,4 -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 permutation trees middle dim 2 height 1 interval

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 2-inter AT-free perfect balanced 2-inter K 1,4 -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 permutation trees middle dim 2 height 1 interval

Inclusion of graph classes 2-inter AT-free perfect balanced 2-inter K 1,4 -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 permutation trees middle dim 2 height 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 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.