Information about On restrictions of balanced 2-interval graphs

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

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 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.

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: 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

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

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)

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

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

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 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

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 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

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.

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. = U G depth 2

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

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

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

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-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

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

(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. 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!

(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

(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 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

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

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 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

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

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

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

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'

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

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!

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatics problems.

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