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 I
5
1
a pair of intervals 8
2
3
4
6
9 7
the pairs of intervals have a non-empty intersection
an edge between two vertices
5
1
G
8 9
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 G C 2 A G G A U U U U C C C A G U A U A U 1 C U G G C C C AC U U C 3
RNA secondary structure prediction U
A
Helices: sets of contiguous base A pairs, appearing successive, or C U nested, in the primary structure. U C I2 I3 I1 G U C I2 C I2 G A successive nested U C U G UUCGU C Find the maximum set of disjoint G successive or nested 2-intervals: G AAGCA dynamic programming. U C UC C G C I1 A A 3 G I A helices C GU G U G G U A U C A A
RNA secondary structure prediction
Pseudo-knot: crossing base pairs. I1
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)
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 5
1
3
4
6 C U C A C G G C 2 A G G A U U U U C C U C A A G A U U 1 C U G G C C C AC U U C 3
8
2 9
7 5
1
8 9
4
2 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 5
1
3
4
6 C U C A C G G C 2 A G G A U U U U C C U C A A G A U U 1 C U G G C C C AC U U C 3
8
2 9
7 5
1
8 9
4
2 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
perfect
AT-free
K1,4-free
circle
Ko sto ch ka claw-free ,W es t, 1 99 9
circ-arc odd-anti cycle-free
co-compar
compar chordal
trapezoid
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
Following ISGCI
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
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 balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
compar
claw-free chordal trapezoid circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
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 B1
I2 B2
B3
l (I 2) < l (I 1)
B4
l (I 3) < l (I 2)
l (I 3) < l (I 1)
I3 B5
B6
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.
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:
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:
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
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
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.
G
depth 2
=
U
Recognition of balanced 2-interval graphs Recognizing balanced 2-interval graphs is NP-complete. G'
v0 v1 z M(v1)
M(v0) H3
H1
H2
Inclusion of graph classes 2-inter balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
compar
claw-free chordal trapezoid circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
Inclusion of graph classes 2-inter balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
compar
claw-free chordal trapezoid circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
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 balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
compar
claw-free chordal trapezoid circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
Inclusion of graph classes 2-inter balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
unit-2-inter
compar
claw-free chordal trapezoid
(2,2)-inter
circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
(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. I1 I2 I3 I4
I5 I6 I7
I8 I5 I6
I8
K4,4-e has a (2,2)-interval realization!
I1 I2 3 4 I I
I7
(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.
a
vl1
X1
vr1 a
X2
G4 X2
vl1
1 r
v2 v'2
v
vl4
vr4
v3 v'3
X4
v4
vr2
X3
vr3
v
vr2
v
vr3
2 l 3 l
X3
v'4
v'1 v'2 v'3 v'4
vl3 vl2
v'1
X1
b v1 v2 v3 v4
v1
X4 vl4 b
vr4
(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.
Inclusion of graph classes 2-inter balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
unit-2-inter
compar
claw-free chordal trapezoid
(2,2)-inter
circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
Inclusion of graph classes 2-inter balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
unit-2-inter
compar
claw-free chordal trapezoid
(2,2)-inter
circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
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 = unit
Proper circular-arc and unit 2-interval graphs Proper circular-arc graphs are unit 2-interval graphs Proof:
+ disjoint intervals
Inclusion of graph classes 2-inter balanced 2-inter
perfect
AT-free
K1,4-free
circle co-compar
unit-2-inter
compar
claw-free chordal trapezoid
(2,2)-inter
circ-arc odd-anti cycle-free
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
Inclusion of graph classes 2-inter balanced 2-inter
Quasi-line graphs: every vertex is AT-free bisimplicial (its neighborhood can be partitioned into 2 cliques). K1,4-free
perfect
circle co-compar
unit-2-inter
claw-free
compar chordal
trapezoid circ-arc
(2,2)-inter
odd-anti cycle-free
quasi-line
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
Inclusion of graph classes 2-inter balanced 2-inter
Quasi-line graphs: every vertex is AT-free bisimplicial (its neighborhood can be partitioned into 2 cliques). K1,4-free
perfect
circle co-compar
unit-2-inter
claw-free
compar chordal
trapezoid circ-arc
(2,2)-inter
odd-anti cycle-free
quasi-line
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
Inclusion of graph classes 2-inter balanced 2-inter all-4-simp
K1,5-free
perfect
AT-free
K1,4-free
circle co-compar
unit-2-inter
claw-free
compar chordal
trapezoid circ-arc
(2,2)-inter
odd-anti cycle-free
quasi-line
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
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,
G
G'
where G' is the complement graph of G + 1 universal vertex
Inclusion of graph classes 2-inter balanced 2-inter all-4-simp
K1,5-free
perfect
AT-free
K1,4-free
circle co-compar
unit-2-inter
claw-free
compar chordal
trapezoid circ-arc
(2,2)-inter
odd-anti cycle-free
quasi-line
outerplanar bipartite
proper circ-arc = circ. interval line unit circ-arc middle
interval
unit = proper interval
co-comp int. dim 2 height 1
permutation
trees
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.
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!
