Permutation graphs an introduction

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Gwendoline Ramsey
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1 Permutation graphs an introduction Ioan Todinca LIFO - Université d Orléans Algorithms and permutations, february /

2 Permutation graphs Optimisation algorithms use, as input, the intersection model (realizer) Recognition algorithms output the intersection(s) model(s) /

3 Plan of the talk. Relationship with other graph classes. Optimisation problems : MaxIndependentSet/MaxClique/Coloring ; Treewidth. Recognition algorithm. Encoding all realizers via modular decomposition. Conclusion /

4 Definition and basic properties Realizer : (π, π ) One can reverse the realizer upside-down or right-left : (π, π ) (π, π ) (π, π ) (π, π ) Complements of permutation graphs are permutation graphs. Reverse the ordering of the bottoms of the segments : (π, π ) (π, π ) /

5 Definition and basic properties Realizer : (π, π ) One can reverse the realizer upside-down or right-left : (π, π ) (π, π ) (π, π ) (π, π ) Complements of permutation graphs are permutation graphs. Reverse the ordering of the bottoms of the segments : (π, π ) (π, π ) /

6 Definition and basic properties Realizer : (π, π ) One can reverse the realizer upside-down or right-left : (π, π ) (π, π ) (π, π ) (π, π ) Complements of permutation graphs are permutation graphs. Reverse the ordering of the bottoms of the segments : (π, π ) (π, π ) /

7 Definition and basic properties Realizer : (π, π ) One can reverse the realizer upside-down or right-left : (π, π ) (π, π ) (π, π ) (π, π ) Complements of permutation graphs are permutation graphs. Reverse the ordering of the bottoms of the segments : (π, π ) (π, π ) /

8 More intersection graph classes Circle graphs Trapezoid graphs Books on graph classes : [Golumbic 80 ; Brandstädt, Le, Spinrad 99 ; Spinrad 00] /

9 MaxIndependentSet via Dynamic Programming Dynamic programming from left to right : MIS[i] = + max MIS[j] j left to i MaxIndependentSet corresponds to the longest increasing sequence in a permutation O(n log n) MaxClique : longest decreasing sequence Coloring : chromatic number = max clique (perfect graphs) /

10 Treewidth via dynamic programming on scanlines Minimal separators correspond to scanlines Bags correspond to areas between two scanlines Treewidth can be solved in polynomial time [Bodlaender, Kloks, Kratsch 9 ; Meister 0] 7/

11 Treewidth via dynamic programming on scanlines Minimal separators correspond to scanlines Bags correspond to areas between two scanlines Treewidth can be solved in polynomial time [Bodlaender, Kloks, Kratsch 9 ; Meister 0] 7/

12 Treewidth via dynamic programming on scanlines Minimal separators correspond to scanlines Bags correspond to areas between two scanlines Treewidth can be solved in polynomial time [Bodlaender, Kloks, Kratsch 9 ; Meister 0] 7/

13 Recognition algorithm Theorem ([Pnueli, Lempel, Even 7], see also [Golumbic 80]) G is a permutation graph if and only if G and G are comparability graphs. Algorithm. Find a transitive orientation of G and one of G. Construct an intersection model for G In O(n + m) time by [McConnell, Spinrad 99] 8/

14 permutation comparability co-comparability Transitive orientation of a permutation graph G : orient edges according to the top endpoints of the segments. If xy, yz E and π (x) < π (y) < π (z) then xz E. 9/

15 permutation comparability co-comparability Transitive orientation of a permutation graph G : orient edges according to the top endpoints of the segments. If xy, yz E and π (x) < π (y) < π (z) then xz E. 9/

16 comparability co-comparability permutation Let E tr be a transitive orientation of G and F tr a transitive orientation of its complement. Lemma E tr F tr induces a total ordering π (E tr F tr ) on the vertex set. 0/

17 comparability co-comparability permutation Let E tr be a transitive orientation of G and F tr a transitive orientation of its complement. Lemma E tr F tr induces a total ordering π (E tr F tr ) on the vertex set. rev(e tr ) F tr induces another total ordering π (rev(e tr ) F tr ). 0/

18 A realizer of G Permutations π (E tr F tr ) and π (rev(e tr ) F tr ) form a realizer of G. Segments x and y intersect iff (xy) E tr and (yx) rev(e tr ) or vice-versa ; equivalently, iff xy E. /

19 Modules and common intervals Substituting a segment (vertex) by the realizer of a permutation graph (module) produces a new permutation graph. A common interval of π and π forms a module in G Strong modules correspond exactly to strong common intervals [de Mongolfier 00] A graph is a permutation graphs iff all prime nodes in the modular decomposition are permutation graphs. /

20 Modules and common intervals Substituting a segment (vertex) by the realizer of a permutation graph (module) produces a new permutation graph. A common interval of π and π forms a module in G Strong modules correspond exactly to strong common intervals [de Mongolfier 00] A graph is a permutation graphs iff all prime nodes in the modular decomposition are permutation graphs. /

21 Encoding realizers Theorem ([Gallai 7]) A prime permutation graph has a unique realizer, up to reversals. The modular decomposition tree + realizers of prime nodes encode all possible realizers of G, cf. [Crespelle, Paul 0]. series parallel prime /

22 Conclusion Summary Many optimization problems become polynomial on permutation graphs Representations (intersection models) based on modular decompositions Some questions Is Bandwidth polynomial or NP-complete on permutation graphs? What about subgraph isomorphism from parametrized point of view? /

23 Conclusion Summary Many optimization problems become polynomial on permutation graphs Representations (intersection models) based on modular decompositions Some questions Is Bandwidth polynomial or NP-complete on permutation graphs? What about subgraph isomorphism from parametrized point of view? Thank you! Your questions? /

Some algorithmic and combinatorial problems on permutation classes

Some algorithmic and combinatorial problems on permutation classes The point of view of decomposition trees PhD Defense, 2009 December the 4th Outline 1 Objects studied : Permutations, Patterns and Classes