permutation graphs --- an introductionigm.univ-mlv.fr/algob/algoperm2012/05todinca.pdfpermutation...

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Introduction Optimization problems Recognition Encoding all realizers Conclusion

Permutation graphs — an introduction

Ioan Todinca

LIFO - Universite d’Orleans

Algorithms and permutations, february 2012

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Permutation graphs

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• Optimisation algorithmsuse, as input, the intersection model (realizer)

• Recognition algorithmsoutput the intersection(s) model(s)

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Plan of the talk

1. Relationship with other graph classes

2. Optimisation problems :MaxIndependentSet/MaxClique/Coloring ;Treewidth

3. Recognition algorithm

4. Encoding all realizers via modular decomposition

5. Conclusion

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Definition and basic properties

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• Realizer : (π1, π2)

• One can ”reverse” the realizer upside-down or right-left :(π1, π2) ∼ (π2, π1) ∼ (π1, π2) ∼ (π2, π1)

• Complements of permutation graphs are permutation graphs.→ Reverse the ordering of the bottoms of the segments :(π1, π2)→ (π1, π2)

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More intersection graph classes

Circle graphs Trapezoid graphs

Books on graph classes : [Golumbic ’80 ; Brandstadt, Le, Spinrad’99 ; Spinrad 2003]

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MaxIndependentSet via Dynamic Programming

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• Dynamic programming from left to right :

MIS [i ] = 1 + maxj left to i

MIS [j ]

• MaxIndependentSet corresponds to the longest increasingsequence in a permutation — O(n log n)

• MaxClique : longest decreasing sequence

• Coloring : chromatic number = max clique (perfect graphs)

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Treewidth via dynamic programming on scanlines

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• Minimal separators correspond to scanlines

• Bags correspond to areas between two scanlines

• Treewidth can be solved in polynomial time [Bodlaender,Kloks, Kratsch 95 ; Meister 2010]

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Recognition algorithm

Theorem ([Pnueli, Lempel, Even ’71], see also [Golumbic ’80])

G is a permutation graph if and only if G and G are comparabilitygraphs.

Algorithm

1. Find a transitive orientation of G and one of G

2. Construct an intersection model for G

In O(n + m) time by [McConnell, Spinrad ’99]

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permutation ⊆ comparability ∩ co-comparability

Transitive orientation of a permutation graph G : orient edgesaccording to the top endpoints of the segments.

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If xy , yz ∈ E and π1(x) < π1(y) < π1(z) then xz ∈ E .

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permutation ⊆ comparability ∩ co-comparability

Transitive orientation of a permutation graph G : orient edgesaccording to the top endpoints of the segments.

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If xy , yz ∈ E and π1(x) < π1(y) < π1(z) then xz ∈ E .

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comparability ∩ co-comparability ⊆ permutation

Let Etr be a transitive orientation of G and Ftr a transitiveorientation of its complement.

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Lemma

Etr ∪ Ftr induces a total ordering π1(Etr ∪ Ftr ) on the vertex set.

rev(Etr ) ∪ Ftr induces another total ordering π2(rev(Etr ) ∪ Ftr ).

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comparability ∩ co-comparability ⊆ permutation

Let Etr be a transitive orientation of G and Ftr a transitiveorientation of its complement.

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Lemma

Etr ∪ Ftr induces a total ordering π1(Etr ∪ Ftr ) on the vertex set.rev(Etr ) ∪ Ftr induces another total ordering π2(rev(Etr ) ∪ Ftr ).

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A realizer of G

Permutations π1(Etr ∪ Ftr ) and π2(rev(Etr ) ∪ Ftr ) form a realizerof G .

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Segments x and y intersect iff (xy) ∈ Etr and (yx) ∈ rev(Etr ) orvice-versa ; equivalently, iff xy ∈ E .

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Modules and common intervals

• Substituting a segment (vertex) by the realizer of apermutation graph (module) produces a new permutationgraph.

• A common interval of π1 and π2 forms a module in G• Strong modules correspond exactly to strong common

intervals [de Mongolfier 2003]

• A graph is a permutation graphs iff all prime nodes in themodular decomposition are permutation graphs.

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Encoding realizers

Theorem ([Gallai ’67])

A prime permutation graph has aunique realizer, up to reversals.

The modular decomposition tree+ realizers of prime nodesencode all possible realizers of G ,cf. [Crespelle, Paul 2010].

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Conclusion

Summary

• Many optimization problems become polynomial onpermutation graphs

• Representations (intersection models) based on modulardecompositions

Some questions

• Is Bandwidth polynomial or NP-complete on permutationgraphs ?

• What about subgraph isomorphism from parametrized pointof view ?

Thank you ! Your questions ?

permutation graphs --- an introductionigm.univ-mlv.fr/algob/algoperm2012/05todinca.pdfpermutation...

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