p-Box: A “new” graph model

Mauricio Soto, Christopher Thraves

DIM, Universidad de Chile LAAS, Toulouse

March 17, 2015

LIAFA, Paris

Wireless Sensor Networks

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Wireless Sensor Networks

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Related Graphs Classes

• Intersection Graphs: G = (V ,E )

V → F (Interval, Disk, Boxes)u 7→ Fu

uv ∈ E ⇔ Fu ∩ Fv = ∅

•

Max-

Tolerance Graphs:

u 7→ (Iu, tu)uv ∈ E ⇔ |Iu ∩ Iv | >= min{tu, tv}.

Related Graphs Classes

• Intersection Graphs: G = (V ,E )

V → F (Interval, Disk, Boxes)u 7→ Fu

uv ∈ E ⇔ Fu ∩ Fv = ∅

• Max-Tolerance Graphs:

u 7→ (Iu, tu)uv ∈ E ⇔ |Iu ∩ Iv | >= max{tu, tv}.

The Model

• p-Box (1) :

v 7→ (Iv , pv )uv ∈ E ⇔ (pv ∈ Bu) ∧ (pu ∈ Bv )

• c-p-Box:

p-Box model where pv is the center of its interval Iv .

The set {(Iv , pv )}v∈V is called (c-)p-Box (1) realization of G .

The Model

• p-Box (1) :

v 7→ (Iv , pv )uv ∈ E ⇔ (pv ∈ Bu) ∧ (pu ∈ Bv )

• c-p-Box:

p-Box model where pv is the center of its interval Iv .

The set {(Iv , pv )}v∈V is called (c-)p-Box (1) realization of G .

The Model

• p-Box (1) :

v 7→ (Iv , pv )uv ∈ E ⇔ (pv ∈ Bu) ∧ (pu ∈ Bv )

• c-p-Box:

p-Box model where pv is the center of its interval Iv .

The set {(Iv , pv )}v∈V is called (c-)p-Box (1) realization of G .

Today

..Boxicity(2) .p-Box(1) . c-p-Box(1).

Interval

.

Outerplanar

. Block.

Max-Tolerance

.

Rooted directed path

.

2DORG

.

Cyclic segment

Related Work

• H. Maehara [’84]: Interval catch digraph vu ∈ D ⇔ pv ∈ Iu

• V, Chepoi, S. Felsner [’13]: Separable rectangle graphs

• J.R. Correa, L. Feuilloley, P. Perez-Lantero, J.A. Soto[’13,LATIN’14]: Diagonal-corner-separated

• D. Catanzaro, S. Chaplick, S. Felsner, B. V. Halldrsson, M.M. Halldrsson, T. Hixon, J. Stacho [’14]: Hook Graph,Point-Tolerance Graphs

.

.

Point-Tolerance= p− Box(1) ⊊ DCS ⊊ SRG

Related Work

• H. Maehara [’84]: Interval catch digraph vu ∈ D ⇔ pv ∈ Iu

• V, Chepoi, S. Felsner [’13]: Separable rectangle graphs

• J.R. Correa, L. Feuilloley, P. Perez-Lantero, J.A. Soto[’13,LATIN’14]: Diagonal-corner-separated

• D. Catanzaro, S. Chaplick, S. Felsner, B. V. Halldrsson, M.M. Halldrsson, T. Hixon, J. Stacho [’14]: Hook Graph,Point-Tolerance Graphs

.

.

Point-Tolerance= p− Box(1) ⊊ DCS ⊊ SRG

Related Work

• H. Maehara [’84]: Interval catch digraph vu ∈ D ⇔ pv ∈ Iu

• V, Chepoi, S. Felsner [’13]: Separable rectangle graphs

• J.R. Correa, L. Feuilloley, P. Perez-Lantero, J.A. Soto[’13,LATIN’14]: Diagonal-corner-separated

• D. Catanzaro, S. Chaplick, S. Felsner, B. V. Halldrsson, M.M. Halldrsson, T. Hixon, J. Stacho [’14]: Hook Graph,Point-Tolerance Graphs

.

.

Point-Tolerance= p− Box(1) ⊊ DCS ⊊ SRG

Related Work

• H. Maehara [’84]: Interval catch digraph vu ∈ D ⇔ pv ∈ Iu

• V, Chepoi, S. Felsner [’13]: Separable rectangle graphs

• J.R. Correa, L. Feuilloley, P. Perez-Lantero, J.A. Soto[’13,LATIN’14]: Diagonal-corner-separated

• D. Catanzaro, S. Chaplick, S. Felsner, B. V. Halldrsson, M.M. Halldrsson, T. Hixon, J. Stacho [’14]: Hook Graph,Point-Tolerance Graphs

.

.

Point-Tolerance= p− Box(1) ⊊ DCS ⊊ SRG

Related Work

• H. Maehara [’84]: Interval catch digraph vu ∈ D ⇔ pv ∈ Iu

• V, Chepoi, S. Felsner [’13]: Separable rectangle graphs

• J.R. Correa, L. Feuilloley, P. Perez-Lantero, J.A. Soto[’13,LATIN’14]: Diagonal-corner-separated

• D. Catanzaro, S. Chaplick, S. Felsner, B. V. Halldrsson, M.M. Halldrsson, T. Hixon, J. Stacho [’14]: Hook Graph,Point-Tolerance Graphs

.

.

Point-Tolerance= p− Box(1) ⊊ DCS ⊊ SRG

Related Work

• Max-Weighted-Independent-Set

▶ NP-complete for Box(2)▶ Linear for Interval Graph (Chordal Graph)▶ O(n3) Hixon[’13](rediscover Lubiw[’91])▶ O(n2) Correa et al[’13] but NP-complete for rectangles

touching diagonal

• Hixon[’13]: χ(G ) is NP-complete.

• Correa et al[’13]: MHS/MIS ∈ [3/2, 2]. (Wegner conjecture)

Related Work

• Max-Weighted-Independent-Set▶ NP-complete for Box(2)

▶ Linear for Interval Graph (Chordal Graph)▶ O(n3) Hixon[’13](rediscover Lubiw[’91])▶ O(n2) Correa et al[’13] but NP-complete for rectangles

touching diagonal

• Hixon[’13]: χ(G ) is NP-complete.

• Correa et al[’13]: MHS/MIS ∈ [3/2, 2]. (Wegner conjecture)

Related Work

• Max-Weighted-Independent-Set▶ NP-complete for Box(2)▶ Linear for Interval Graph (Chordal Graph)

▶ O(n3) Hixon[’13](rediscover Lubiw[’91])▶ O(n2) Correa et al[’13] but NP-complete for rectangles

touching diagonal

• Hixon[’13]: χ(G ) is NP-complete.

• Correa et al[’13]: MHS/MIS ∈ [3/2, 2]. (Wegner conjecture)

Related Work

• Max-Weighted-Independent-Set▶ NP-complete for Box(2)▶ Linear for Interval Graph (Chordal Graph)▶ O(n3) Hixon[’13](rediscover Lubiw[’91])

▶ O(n2) Correa et al[’13] but NP-complete for rectanglestouching diagonal

• Hixon[’13]: χ(G ) is NP-complete.

• Correa et al[’13]: MHS/MIS ∈ [3/2, 2]. (Wegner conjecture)

Related Work

• Max-Weighted-Independent-Set▶ NP-complete for Box(2)▶ Linear for Interval Graph (Chordal Graph)▶ O(n3) Hixon[’13](rediscover Lubiw[’91])▶ O(n2) Correa et al[’13] but NP-complete for rectangles

touching diagonal

• Hixon[’13]: χ(G ) is NP-complete.

• Correa et al[’13]: MHS/MIS ∈ [3/2, 2]. (Wegner conjecture)

Related Work

• Max-Weighted-Independent-Set▶ NP-complete for Box(2)▶ Linear for Interval Graph (Chordal Graph)▶ O(n3) Hixon[’13](rediscover Lubiw[’91])▶ O(n2) Correa et al[’13] but NP-complete for rectangles

touching diagonal

• Hixon[’13]: χ(G ) is NP-complete.

• Correa et al[’13]: MHS/MIS ∈ [3/2, 2]. (Wegner conjecture)

Related Work

• Max-Weighted-Independent-Set▶ NP-complete for Box(2)▶ Linear for Interval Graph (Chordal Graph)▶ O(n3) Hixon[’13](rediscover Lubiw[’91])▶ O(n2) Correa et al[’13] but NP-complete for rectangles

touching diagonal

• Hixon[’13]: χ(G ) is NP-complete.

• Correa et al[’13]: MHS/MIS ∈ [3/2, 2]. (Wegner conjecture)

Intersection Model

..1.

3

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4

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B1

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B2

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•.

B4

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(p1,−p1)

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(p2,−p2)

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(p3,−p3)

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(p4,−p4)

• Kaufmann et al.[SODA’06]: Max-tolerance = Intersectionof isosceles triangles

Max-tolerance ⊃ c-p-Box(1).

Intersection Model

..1.

3

. 2.

4

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•

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B1

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•.

B2

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•.

B4

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(p1,−p1)

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(p2,−p2)

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(p3,−p3)

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(p4,−p4)

• Kaufmann et al.[SODA’06]: Max-tolerance = Intersectionof isosceles triangles

Max-tolerance ⊃ c-p-Box(1).

Intersection Model

..1.

3

. 2.

4

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•

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B1

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B2

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B4

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(p1,−p1)

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(p2,−p2)

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(p3,−p3)

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(p4,−p4)

• Kaufmann et al.[SODA’06]: Max-tolerance = Intersectionof isosceles triangles

Max-tolerance ⊃ c-p-Box(1).

Intersection Model

..1.

3

. 2.

4

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•

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B1

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B2

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B4

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(p1,−p1)

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(p2,−p2)

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(p3,−p3)

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(p4,−p4)

• Kaufmann et al.[SODA’06]: Max-tolerance = Intersectionof isosceles triangles

Max-tolerance ⊃ c-p-Box(1).

Intersection Model

..1.

3

. 2.

4

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•

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B1

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B2

. •. B3.

•.

B4

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(p1,−p1)

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(p2,−p2)

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(p3,−p3)

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(p4,−p4)

• Kaufmann et al.[SODA’06]: Max-tolerance = Intersectionof isosceles triangles

Max-tolerance ⊃ c-p-Box(1).

Intersection Model

..1.

3

. 2.

4

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•

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B1

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B2

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B4

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(p1,−p1)

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(p2,−p2)

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(p3,−p3)

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(p4,−p4)

• Kaufmann et al.[SODA’06]: Max-tolerance = Intersectionof isosceles triangles

Max-tolerance ⊃ c-p-Box(1).

Combinatorial Characterization

Positions of representative points pv induce an order of V .

..•.x

. •.

u

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v

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y

. ⇒. •.

x

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u

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• Interval graphs : [S. Olariu ’91]

..•.x

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u

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v

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x

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u

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v

• Outerplanar : [T. Bilski ’92] Page number 1.

We will prove: {Interval, Outerplanar}∈ c-p-Box (1)

Combinatorial Characterization

Positions of representative points pv induce an order of V .

..•.x

. •.

u

. •.

v

. •.

y

. ⇒. •.

x

. •.

u

. •.

v

. •.

y

• Interval graphs : [S. Olariu ’91]

..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

• Outerplanar : [T. Bilski ’92] Page number 1.

We will prove: {Interval, Outerplanar}∈ c-p-Box (1)

Combinatorial Characterization

Positions of representative points pv induce an order of V .

..•.x

. •.

u

. •.

v

. •.

y

. ⇒. •.

x

. •.

u

. •.

v

. •.

y

• Interval graphs : [S. Olariu ’91]

..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

• Outerplanar : [T. Bilski ’92] Page number 1.

We will prove: {Interval, Outerplanar}∈ c-p-Box (1)

Combinatorial Characterization

Positions of representative points pv induce an order of V .

..•.x

. •.

u

. •.

v

. •.

y

. ⇒. •.

x

. •.

u

. •.

v

. •.

y

• Interval graphs : [S. Olariu ’91]

..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

• Outerplanar : [T. Bilski ’92] Page number 1.

We will prove: {Interval, Outerplanar}∈ c-p-Box (1)

Combinatorial Characterization

Positions of representative points pv induce an order of V .

..•.x

. •.

u

. •.

v

. •.

y

. ⇒. •.

x

. •.

u

. •.

v

. •.

y

• Interval graphs : [S. Olariu ’91]

..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

• Outerplanar : [T. Bilski ’92] Page number 1.

We will prove: {Interval, Outerplanar}∈ c-p-Box (1)

Combinatorial Characterization cont.

Rooted directed path graph : intersection graphs of of directedpaths in a rooted directed tree

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b

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c

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z

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An inverse DFS on the tree is (j i h g f e d c b a) inducing the orderof the vertices of the graph: (t z y w v x u)

Combinatorial Characterization cont.

Rooted directed path graph : intersection graphs of of directedpaths in a rooted directed tree

..a.

b

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c

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d

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e

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f

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g

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h

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i

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j

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x

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u

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y

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v

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w

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z

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t

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u

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x

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v

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y

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w

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z

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t

An inverse DFS on the tree is (j i h g f e d c b a) inducing the orderof the vertices of the graph: (t z y w v x u)

Combinatorial Characterization cont.

Rooted directed path graph : intersection graphs of of directedpaths in a rooted directed tree

..a.

b

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c

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d

.

e

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f

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g

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h

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i

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j

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x

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u

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y

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v

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w

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t

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u

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x

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An inverse DFS on the tree is (j i h g f e d c b a) inducing the orderof the vertices of the graph: (t z y w v x u)

The c-p-Box (1) class

Some Definitions

• Given an order π of the vertex, we note by:▶ ℓπ(v) the most left neighbor of v .▶ ρπ(v) the most right neighbor of v .

• Given a realization of G :▶ L(v) denotes the left extreme of Iv▶ R(v) denotes the right extreme of Iv▶ v is safe if its position pv belongs only to its neighbors

intervals.

Some Definitions

• Given an order π of the vertex, we note by:▶ ℓπ(v) the most left neighbor of v .▶ ρπ(v) the most right neighbor of v .

• Given a realization of G :▶ L(v) denotes the left extreme of Iv▶ R(v) denotes the right extreme of Iv▶ v is safe if its position pv belongs only to its neighbors

intervals.

Interval Graphs and c-p-Box (1)

Theorem

Interval ⊂ c-p-Box (1).

[S. Olariu 1991]..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

We greedily construct a realization according to order s.t. at step i :1. pk−1 < pk

2. ρ(j) <π ρ(k) ⇒ R(j) <R(k)

3. L(j) < pℓ(j)

4. ρ(k) <π j ⇔ R(k) < pj

First, set position pi after pi1 and s.t. is contained only byintervals associated to its previous neighbors.

Second, set the interval Ii s.t. it contains all its previousneighbors.

Finally, we modify, if necessary, the interval of previousvertices in order to satisfy conditions 2.

All vertices are safe with respect to previous neighbors.

Interval Graphs and c-p-Box (1)

Theorem

Interval ⊂ c-p-Box (1).

[S. Olariu 1991]..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

We greedily construct a realization according to order s.t. at step i :1. pk−1 < pk

2. ρ(j) <π ρ(k) ⇒ R(j) <R(k)

3. L(j) < pℓ(j)

4. ρ(k) <π j ⇔ R(k) < pj

First, set position pi after pi1 and s.t. is contained only byintervals associated to its previous neighbors.

Second, set the interval Ii s.t. it contains all its previousneighbors.

Finally, we modify, if necessary, the interval of previousvertices in order to satisfy conditions 2.

All vertices are safe with respect to previous neighbors.

Interval Graphs and c-p-Box (1)

Theorem

Interval ⊂ c-p-Box (1).

[S. Olariu 1991]..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

We greedily construct a realization according to order s.t. at step i :1. pk−1 < pk

2. ρ(j) <π ρ(k) ⇒ R(j) <R(k)

3. L(j) < pℓ(j)

4. ρ(k) <π j ⇔ R(k) < pj

First, set position pi after pi1 and s.t. is contained only byintervals associated to its previous neighbors.

Second, set the interval Ii s.t. it contains all its previousneighbors.

Finally, we modify, if necessary, the interval of previousvertices in order to satisfy conditions 2.

All vertices are safe with respect to previous neighbors.

Interval Graphs and c-p-Box (1)

Theorem

Interval ⊂ c-p-Box (1).

[S. Olariu 1991]..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

We greedily construct a realization according to order s.t. at step i :1. pk−1 < pk

2. ρ(j) <π ρ(k) ⇒ R(j) <R(k)

3. L(j) < pℓ(j)

4. ρ(k) <π j ⇔ R(k) < pj

First, set position pi after pi1 and s.t. is contained only byintervals associated to its previous neighbors.

Second, set the interval Ii s.t. it contains all its previousneighbors.

Finally, we modify, if necessary, the interval of previousvertices in order to satisfy conditions 2.

All vertices are safe with respect to previous neighbors.

Interval Graphs and c-p-Box (1)

Theorem

Interval ⊂ c-p-Box (1).

[S. Olariu 1991]..•.x

. •.

u

. •.

v

. ⇒. •.

x

. •.

u

. •.

v

We greedily construct a realization according to order s.t. at step i :1. pk−1 < pk

2. ρ(j) <π ρ(k) ⇒ R(j) <R(k)

3. L(j) < pℓ(j)

4. ρ(k) <π j ⇔ R(k) < pj

First, set position pi after pi1 and s.t. is contained only byintervals associated to its previous neighbors.

Second, set the interval Ii s.t. it contains all its previousneighbors.

Finally, we modify, if necessary, the interval of previousvertices in order to satisfy conditions 2.

All vertices are safe with respect to previous neighbors.

Outerplanar Graphs and c-p-Box (1)

Theorem

Outerplanar ⊂ c-p-Box (1).

• Non trivial biconnected components are dissections ofpolygons.

.. 1.

2

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3

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4

.5 .

6

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7

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8

• Cycles are in c-p-Box (1)

• How to “glue” two cycles by an edge

• How to “glue” two biconnected components by a vertex

Outerplanar and c-p-Box (1)

Cn ∈ c-p-Box(1). Moreover, if π denotes the permutation inducedby a realization Then, there exists a clockwise (or anticlockwise)labeling l : V → {1, 2, . . . , n} such that:

1. π(l−1(1)) = 1 ∧ π(l−1(n)) = n.

2. p-Box (1): ∀u ∈ V , |l(u)− π(u)| ≤ 1.

3. c-p-Box (1): ∀u ∈ V , l(u) = π(u).

Extremes vertices are safe!

Outerplanar and c-p-Box (1)

Cn ∈ c-p-Box(1). Moreover, if π denotes the permutation inducedby a realization Then, there exists a clockwise (or anticlockwise)labeling l : V → {1, 2, . . . , n} such that:

1. π(l−1(1)) = 1 ∧ π(l−1(n)) = n.

2. p-Box (1): ∀u ∈ V , |l(u)− π(u)| ≤ 1.

3. c-p-Box (1): ∀u ∈ V , l(u) = π(u).

Extremes vertices are safe!

Outerplanar and c-p-Box (1)

Cn ∈ c-p-Box(1). Moreover, if π denotes the permutation inducedby a realization Then, there exists a clockwise (or anticlockwise)labeling l : V → {1, 2, . . . , n} such that:

1. π(l−1(1)) = 1 ∧ π(l−1(n)) = n.

2. p-Box (1): ∀u ∈ V , |l(u)− π(u)| ≤ 1.

3. c-p-Box (1): ∀u ∈ V , l(u) = π(u).

Extremes vertices are safe!

Outerplanar and c-p-Box (1)

Cn ∈ c-p-Box(1). Moreover, if π denotes the permutation inducedby a realization Then, there exists a clockwise (or anticlockwise)labeling l : V → {1, 2, . . . , n} such that:

1. π(l−1(1)) = 1 ∧ π(l−1(n)) = n.

2. p-Box (1): ∀u ∈ V , |l(u)− π(u)| ≤ 1.

3. c-p-Box (1): ∀u ∈ V , l(u) = π(u).

Extremes vertices are safe!

Outerplanar and c-p-Box (1)

Cn ∈ c-p-Box(1). Moreover, if π denotes the permutation inducedby a realization Then, there exists a clockwise (or anticlockwise)labeling l : V → {1, 2, . . . , n} such that:

1. π(l−1(1)) = 1 ∧ π(l−1(n)) = n.

2. p-Box (1): ∀u ∈ V , |l(u)− π(u)| ≤ 1.

3. c-p-Box (1): ∀u ∈ V , l(u) = π(u).

Extremes vertices are safe!

Outerplanar and c-p-Box (1)

We can “glue” cycles by an edge in A DFS of weak dual

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Outerplanar and c-p-Box (1)

We construct a realization according to a BFS on the Block-tree ofG scaling biconnected component

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p-Box (1)∖ c-p-Box (1)

..•.a . •.x1

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y1

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yly−1

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.

•

.

zlz−1

..

H lx ,ly ,lz

. •. a... •.x1

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H2,ly ,lz

. •. a. •.x1

. •.x2

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H3,ly ,lz

..•.a. •.

z1. •.

z2. •.

zlz−1

. •.

b

. •.

x. •.

ylv−1

. •.

y1. •.

y1. •.

x1. •.

a. •.

z1. •.

zlz−1

. •.

b

. •.

x2. •.

ylv−1

. •.

y2

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx ≥ 2 does not belongto c-p-Box (1).

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx > 3 does not belongto p-Box (1).

..•.a . •.x1

. •.xlx−1

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H lx ,ly ,lz

. •. a... •.x1

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H2,ly ,lz

. •. a. •.x1

. •.x2

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H3,ly ,lz

..•.a. •.

z1. •.

z2. •.

zlz−1

. •.

b

. •.

x. •.

ylv−1

. •.

y1. •.

y1. •.

x1. •.

a. •.

z1. •.

zlz−1

. •.

b

. •.

x2. •.

ylv−1

. •.

y2

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx ≥ 2 does not belongto c-p-Box (1).

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx > 3 does not belongto p-Box (1).

..•.a . •.x1

. •.xlx−1

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H lx ,ly ,lz

. •. a... •.x1

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H2,ly ,lz

. •. a. •.x1

. •.x2

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H3,ly ,lz

..•.a. •.

z1. •.

z2. •.

zlz−1

. •.

b

. •.

x. •.

ylv−1

. •.

y1. •.

y1. •.

x1. •.

a. •.

z1. •.

zlz−1

. •.

b

. •.

x2. •.

ylv−1

. •.

y2

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx ≥ 2 does not belongto c-p-Box (1).

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx > 3 does not belongto p-Box (1).

..•.a . •.x1

. •.xlx−1

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H lx ,ly ,lz

. •. a... •.x1

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H2,ly ,lz

. •. a. •.x1

. •.x2

. •. b.

•

.

y1

.

•

.

yly−1

.

•

.

z1

.

•

.

zlz−1

..

H3,ly ,lz

..•.a. •.

z1. •.

z2. •.

zlz−1

. •.

b

. •.

x. •.

ylv−1

. •.

y1. •.

y1. •.

x1. •.

a. •.

z1. •.

zlz−1

. •.

b

. •.

x2. •.

ylv−1

. •.

y2

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx ≥ 2 does not belongto c-p-Box (1).

• Any H lx ,ly ,lz graph such that lz ≥ ly ≥ lx > 3 does not belongto p-Box (1).

Future Work and Open Questions

• Combinatorial characterization for c-p-Box (1).

▶ Given an order?

• Recognition complexity of (c-)p-Box (1).

• Related graph classes

▶ Unit-p-Box▶ p-Box with arcs

Future Work and Open Questions

• Combinatorial characterization for c-p-Box (1).▶ Given an order?

• Recognition complexity of (c-)p-Box (1).

• Related graph classes

▶ Unit-p-Box▶ p-Box with arcs

Future Work and Open Questions

• Combinatorial characterization for c-p-Box (1).▶ Given an order?

• Recognition complexity of (c-)p-Box (1).

• Related graph classes

▶ Unit-p-Box▶ p-Box with arcs

Future Work and Open Questions

• Combinatorial characterization for c-p-Box (1).▶ Given an order?

• Recognition complexity of (c-)p-Box (1).

• Related graph classes

▶ Unit-p-Box▶ p-Box with arcs

Future Work and Open Questions

• Combinatorial characterization for c-p-Box (1).▶ Given an order?

• Recognition complexity of (c-)p-Box (1).

• Related graph classes▶ Unit-p-Box

▶ p-Box with arcs

Future Work and Open Questions

• Combinatorial characterization for c-p-Box (1).▶ Given an order?

• Recognition complexity of (c-)p-Box (1).

• Related graph classes▶ Unit-p-Box▶ p-Box with arcs

Duality gap p-Box (1), Correa et al. [’13]

• MIS(G ) ≥ max{|Ix |, |Iy |} = max{|Hx |, |Hy |}• MHS(G ) ≤ H ≤ |Hx |+ |Hy | ≤ 2MIS(G )

WMIS in p-Box (1), Cantazaro et al.[’13]

• opt[u, v ] = maxBi∈[a,b]{opt[a,Bi ] + w(Bi ) + opt[i , b]}• O(n2) pairs computed in O(n)

c-p-Box with given order

min∑i∈V

ri (1)

s.t. xi − xℓπ(i) ≤ ri − ε1 1 ≤ i ≤ n (2)

xρπ(i) − xi ≤ ri − ε1 1 ≤ i ≤ n (3)

xj − xi ≥ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ↷ j(4)

xj − xi ≥ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ↶ j(5)

xj − xi ≤ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(6)

xj − xi ≤ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(7)

xi+1 − xi ≥ ε2 1 ≤ i ≤ n − 1 (8)

x1 = 0, xn = L

xi , ri ≥ 0 1 ≤ i ≤ n

c-p-Box with given order

min∑i∈V

ri (1)

s.t. xi − xℓπ(i) ≤ ri − ε1 1 ≤ i ≤ n (2)

xρπ(i) − xi ≤ ri − ε1 1 ≤ i ≤ n (3)

xj − xi ≥ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ↷ j(4)

xj − xi ≥ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ↶ j(5)

xj − xi ≤ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(6)

xj − xi ≤ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(7)

xi+1 − xi ≥ ε2 1 ≤ i ≤ n − 1 (8)

x1 = 0, xn = L

xi , ri ≥ 0 1 ≤ i ≤ n

c-p-Box with given order

min∑i∈V

ri (1)

s.t. xi − xℓπ(i) ≤ ri − ε1 1 ≤ i ≤ n (2)

xρπ(i) − xi ≤ ri − ε1 1 ≤ i ≤ n (3)

xj − xi ≥ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ↷ j(4)

xj − xi ≥ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ↶ j(5)

xj − xi ≤ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(6)

xj − xi ≤ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(7)

xi+1 − xi ≥ ε2 1 ≤ i ≤ n − 1 (8)

x1 = 0, xn = L

xi , ri ≥ 0 1 ≤ i ≤ n

c-p-Box with given order

min∑i∈V

ri (1)

s.t. xi − xℓπ(i) ≤ ri − ε1 1 ≤ i ≤ n (2)

xρπ(i) − xi ≤ ri − ε1 1 ≤ i ≤ n (3)

xj − xi ≥ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ↷ j(4)

xj − xi ≥ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ↶ j(5)

xj − xi ≤ ri 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(6)

xj − xi ≤ rj 1 ≤ i <π j ≤ n, ij /∈ E , i ≬ j(7)

xi+1 − xi ≥ ε2 1 ≤ i ≤ n − 1 (8)

x1 = 0, xn = L

xi , ri ≥ 0 1 ≤ i ≤ n

Containment Relations

..Boxicity(2) .p-Box(1) . c-p-Box(1).

Interval

.

Outerplanar

. Block.

Max-Tolerance

.

Rooted directed path

.

2DORG

.

Cyclic segment