On the structure of (pan, even hole)-free graphs Kathie Cameron 1 , - - PowerPoint PPT Presentation

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On the structure of (pan, even hole)-free graphs Kathie Cameron 1 , - - PowerPoint PPT Presentation

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H-Free Graphs Decomposition Theorem Applications On the structure of (pan, even hole)-free graphs Kathie Cameron 1 , Steven Chaplick 2 , Chnh T. Hong 3 1 Department of Mathematics, Wilfrid Laurier University (Canada) 2 Institut fur

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H-Free Graphs Decomposition Theorem Applications

On the structure of (pan, even hole)-free graphs

Kathie Cameron1, Steven Chaplick2, Chính T. Hoàng3

1Department of Mathematics, Wilfrid Laurier University (Canada) 2Institut fur Mathematik, Technische Universitat Berlin (Germany) 2Department of Physics and Computer Science, Wilfrid Laurier University

(Canada)

June 19, 2015

Adriatic Coast Graph Theory 2015.

Support by GraDR EUROGIGA and NSERC.

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H-Free Graphs Decomposition Theorem Applications

Outline

1

H-Free Graphs

2

Decomposition Theorem

3

Applications

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H-Free Graphs Decomposition Theorem Applications

Outline

1

H-Free Graphs

2

Decomposition Theorem

3

Applications

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H-Free Graphs Decomposition Theorem Applications

H-Free Graphs

Definition For a graph H, a graph G is H-free, when G does not contain H as an induced subgraph.

H H-free Proper Colouring

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H-Free Graphs Decomposition Theorem Applications

Claws, holes, and pans

We will deal with (claw, even hole)-free graphs Even holes: holes with even length (Pan, even hole)-free graphs

  • S. Olariu introduces pan, proved SPGC for pan-free graphs.

Stability number of pan-free graphs is in P (Brandstadt, Lozin, Mosca)

  • claw
  • hole
  • pan
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H-Free Graphs Decomposition Theorem Applications

Related Graph Classes and Recognition

Chordal: G is hole-free; i.e., (C4, C5, . . .)-free: linear time [Rose, Tarjan, Lueker; SIAM JComp 1976] Odd-hole-free; i.e., (C5, C7, . . .)-free: OPEN Perfect: (odd-hole,odd-anti-hole)-free: polytime [Chudnovsky, Cornuejols, Liu, Seymour, Vuškovi´ c; Combinatorica 2005] Even-hole-free; i.e., (C4, C6, . . .)-free: polytime (more on this) Note: Information System on Graph Classes http://www.graphclasses.org/ defines even-hole free as (C6, C8, . . .)-free. C4 is not excluded, but we do exclude C4!!

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H-Free Graphs Decomposition Theorem Applications

Finding Even-Holes

O(n40) [Conforti, Cornuéjols, Kapoor, and Vuškovi´ c; JGT 2002]. O(n31) [Chudnovsky, Kawarabayashi, and Seymour; JGT 2005]. O(n19) [da Silva and Vuškovi´ c; JCTB 2013]. O(m3n5) [Chang and Lu; SODA 2012, arxiv 2013]. In planar: O(n3) [Porto; LATIN 1992] In claw-free: O(n8) [van ’t Hof, Kami´ nski, Paulusma; Algorithmica 2012] In circular-arc: O(mn2loglogn) [Cameron, Eschen, Hoàng, Sritharan; 2007]

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H-Free Graphs Decomposition Theorem Applications

Combinatorial Optimization Problems

Clique

  • Ind. Set

Colouring Clique cover Even-Hole-free P ? ? NP-hard Odd-Hole-free NP-hard P NP-hard ?? Pan-free NP-hard P NP-hard NP-hard (Pan, Even hole)-free P P P ?? Even-hole-free graphs: χ(G) ≤ 2ω(G) − 1 [Addario-Berry, Chudnovsky, Havet, Reed, Seymour; JCTB 2008]

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H-Free Graphs Decomposition Theorem Applications

Our Results

Theorem For a (pan,even-hole)-free graph G, one of the following hold:

1

G is a clique.

2

G contains a clique cutset.

3

G is a unit circular arc graph

4

G is the join of a clique and a unit circular arc graph. Recognition in O(nm) time. Colouring in O(n2.5 + nm) time.

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H-Free Graphs Decomposition Theorem Applications

Circular Arc Graphs

“unit” means all arcs have the same length Colouring is NP-complete for circular arc graphs

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H-Free Graphs Decomposition Theorem Applications

Outline

1

H-Free Graphs

2

Decomposition Theorem

3

Applications

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H-Free Graphs Decomposition Theorem Applications

Our Decomposition Theorem

Theorem For a (pan,even-hole)-free graph G, one of the following hold:

1

G is a clique.

2

G contains a clique cutset.

3

G is an unit circular arc graph.

4

G is the join of a clique and a unit circular arc graph. Auxiliary structure generalizing holes: buoy

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H-Free Graphs Decomposition Theorem Applications

Holes and Buoys

A length ℓ-buoy has ℓ bags: B0, . . . , Bℓ−1, each bag is a clique, and each vertex in a bag has neighbours in adjacent bags (but not other bags).

B1 B2 B3 B4 B0

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H-Free Graphs Decomposition Theorem Applications

Our Decomposition Theorem

Theorem For a (pan,even-hole)-free graph G, one of the following hold:

1

G is a clique.

2

G contains a clique cutset.

3

G is a buoy*

4

G is the join of a clique and a 5-buoy*. * These buoys are extremely special, as we will see.

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H-Free Graphs Decomposition Theorem Applications

Structure of a Buoy

Theorem If B is a ℓ-buoy in a (pan,even-hole)-free graph, then: Each Bi can be ordered by neighbourhood inclusion. Either (Bi ∪ Bi+1) or Bi ∪ Bi−1 is a clique. For efficient recognition: Theorem If B is an ℓ-buoy where each Bi can be ordered by neighbourhood inclusion, then every hole in B has length ℓ.

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H-Free Graphs Decomposition Theorem Applications

Structure of a Buoy

Theorem If B is a ℓ-buoy in a (pan,even-hole)-free graph, then: Each Bi can be ordered by neighbourhood inclusion. Either (Bi ∪ Bi+1) or (Bi ∪ Bi−1) is a clique.

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H-Free Graphs Decomposition Theorem Applications

Buoys To Circular Arcs

Remember: each bag is orderable by neighbourhood inclusion.

Bi Bi+1 x2 b1 x1 b2 bt -1

(i,1) (i,2) (i, ) (i) (i+1)

... ... ... bti xti xt -1

i

(i, -1) ti ti

i

... (0) (1)

( -1)

(i+1) (i)

. . . ... ...

{

{

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H-Free Graphs Decomposition Theorem Applications

Buoys To Unit Circular Arcs

Case: Bi ∪ Bi+1 is not a clique. Remember: every other pair of bags is a clique.

Bi Bi+1 x2 b1 x1 b2 bt -1

(i,1) (i,2) (i, ) (i) (i+1)

... ... ... bti xti xt -1

i

(i, -1) ti ti

i

...

{

Ai+

{

Ai

{

Ai+1

(i,1) (i,2)

...

(i, -1) ti

...

{

length=1

{

length=1

{

Ai-1+

(i,1) (i,2)

...

(i, -1) ti

...

{

length=1

{

length=1

{

Ai+1+

(i, ) ti (i, ) ti

length=1

Bi-1 Bi+2

(i-1) (i+2)

Each arc will have length 2+ǫ Arc Ai has length ǫ

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H-Free Graphs Decomposition Theorem Applications

Neighbourhood of a buoy

Theorem Let B be an ℓ-buoy in a (pan,even-hole)-free graph and let x be a neighbour B. Then: x adjacent to 5 bags implies ℓ = 5 and x universal to B (*). x adjacent to 2 bags implies these bags are consecutive and form a clique. x adjacent to 3 bags implies these bags are consecutive and x universal to the middle bag. (*) These vertices are the only way we have a unit circular arc graph joined with a clique in our decomposition.

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H-Free Graphs Decomposition Theorem Applications

Neighbourhood of a buoy

x adjacent to 3 bags implies these bags are consecutive and x universal to the middle bag.

Bi Bi+1 Bi-1 bi+1 bi ai-1 ai+1 bi-1

x

Bi Bi+1 Bi-1 bi ai-1 ai+1

x

Bi Bi+1 Bi-1 bi+1 bi ai-1 ai+1

x

Bi-2 bi-2

(1) (3.1) (2) di

Bi Bi+1 Bi-1 bi+1 bi ai-1 ai+1

x

Bi-2 bi-2

(3.2) di

Bi+2 di+2

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H-Free Graphs Decomposition Theorem Applications

Decomposition Theorem

Theorem Consider a (pan,even-hole)-free graph G. Let B be a "maximal" buoy of G:

1

B contains all vertices of G

2

G contains a clique cutset.

3

G is the join of a clique and a 5-buoy (unit circular arc graph).

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H-Free Graphs Decomposition Theorem Applications

Structure of Maximal Buoys

A1

U is a clique and has no neighbours

  • utside B

U

Ai R

If A1 not empty, clique cutset

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H-Free Graphs Decomposition Theorem Applications

Structure of Maximal Buoys

A1

U is a clique and has no neighbours

  • utside B

U

Ai R

If A1 not empty, clique cutset

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H-Free Graphs Decomposition Theorem Applications

Outline

1

H-Free Graphs

2

Decomposition Theorem

3

Applications

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H-Free Graphs Decomposition Theorem Applications

Main Tool Clique Cutset Decomposition

K G G1 Gt ... K G1 Gt ... K

Computation in O(nm) time with < n atoms [Tarjan; JDM 1985] Applications: Chromatic number, and the presence of a

  • hole. [Whitesides; 1984]
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H-Free Graphs Decomposition Theorem Applications

Colouring

Note: only need to consider atoms and our atoms are unit circular arc graphs. 1. Run Clique Cutset decomposition : O(nm) time, with < n atoms 2. Colour the atoms of the decomposition: O(n1.5 + m) per atom. 3. Now, χ(G) = max{χ(H) : H is an atom of G}. Total time: O(n2.5 + nm). χ-bounded: χ(G) ≤ 1.5 ω(G). Unit Circular Arc representation construction: O(n + m) [Lin, Szwarcfiter; SIAM JDM 2008] Unit Circular Arc colouring from a representation: O(n1.5) [Shih, Hsu; JDAM 1989]

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H-Free Graphs Decomposition Theorem Applications

Recognition

1. Run Clique Cutset decomposition : O(nm) time, with < n atoms 2. For each atom: 3. verify that no holes of the atom form a pan with a vertex outside 5. Build our special buoy B from this hole: O(n + m). 6. If B cannot be built, we produce a pan or an even hole 7. Build an unit circular arc representation. Total time: O(nm). Chordality Testing: O(n + m) : [Rose, Tarjan, Lueker; SIAM JComp 1976]

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H-Free Graphs Decomposition Theorem Applications

Concluding Remarks

(pan,even-hole)-free graphs decompose into *almost* unit-circular arc graphs by clique cutsets. This allows: Recognition in O(nm + m1.69) time. Colouring in O(n2.5 + nm) time. Bounding parameters : ω(G) ≤ χ(G) ≤ 1.5ω(G). Open Problems: Odd-hole-free: recognition, independent set, structural characterization. Even-hole-free: independent set, colouring. (pan,even-hole)-free: clique cover. Characterize circular arc graphs by minimal forbidden induced subgraphs.