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Fakultt fr Mathematik und Informatik Characterisations of Game-perfect Graphs and Digraphs Dominique Andres (joint work with: Edwin Lock) Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock Fakultt fr Mathematik und

  1. Fakultät für Mathematik und Informatik Game-perfect graphs Let g = [ X , Y ] be one of the 6 games. Definition (game chromatic number) Definition (clique number) χ g ( G ) := smallest number of colours such that ω ( G ) := size of a largest clique Alice has a winning strategy. of G . Definition (game-perfect graph) A graph G is game-perfect (or g -perfect or [ X , Y ] -perfect) if, for any induced subgraph H of G , χ g ( H ) = ω ( H ) . Folie 8 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  2. Fakultät für Mathematik und Informatik Game-perfect graphs Let g = [ X , Y ] be one of the 6 games. Definition (game chromatic number) Definition (clique number) χ g ( G ) := smallest number of colours such that ω ( G ) := size of a largest clique Alice has a winning strategy. of G . Definition (game-perfect graph) A graph G is game-perfect (or g -perfect or [ X , Y ] -perfect) if, for any induced subgraph H of G , χ g ( H ) = ω ( H ) . Remark. Game-perfect graphs are special cases of perfect graphs. Folie 8 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  3. Fakultät für Mathematik und Informatik 6 classes of game-perfect graphs [ B , B ] -perfect graphs: structural characterisation A. 2012 [ A , B ] -perfect graphs: structural characterisation A. 2012 [ A , − ] -perfect graphs: structural characterisation A. 2012 [ B , − ] -perfect graphs: structural characterisation with Edwin Lock 2016+ [ B , A ] -perfect graphs: open problem (no characterisation known) [ A , A ] -perfect graphs: open problem (no characterisation known) Folie 9 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  4. Fakultät für Mathematik und Informatik What does such a characterisation look like? Theorem Let G be a graph. (i) G is game-perfect if and only if (ii) G does not contain any of the forbidden structures F j (left). This is the case if and only if (iii) G belongs to one of the structural types E i (right). forbidden induced subgraphs: allowed structures: F 1 , . . . , F n E 1 , . . . , E m Folie 10 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  5. Fakultät für Mathematik und Informatik What does such a characterisation look like? game [ B , B ] Theorem (A. 2012) Let G be a graph. (i) G is [ B , B ] -perfect if and only if (ii) G does not contain any of the forbidden structures F j (left). This is the case if and only if (iii) G belongs to one of the structural types E i (right). 4 forbidden induced subgraphs: 1 allowed structure: H 0 K q K p K r H 4 H 1 H 3 H 2 × k Folie 11 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  6. Fakultät für Mathematik und Informatik What does such a characterisation look like? game [ B , − ] Theorem (with Edwin Lock 2016+) Let G be a graph. (i) G is [ B , − ] -perfect if and only if (ii) G does not contain any of the forbidden structures F j (left). This is the case if and only if (iii) G belongs to one of the structural types E i (right). 15 forbidden induced subgraphs: 9 allowed structures: H 0 b c K q K p K r K b K a K c K m K n d c b c e H 4 e H 1 x 1 x 2 c d e a b c d a e a a b d d H 3 × k H 2 a b c d e b a a b a K b K d g A 1 V 1 b e c e d A 2 V 2 K m c K n a b c d e e d A R V R K c c d f d a f f A 3 V 3 b d a a A 4 V 4 K a b e f g c e c a c g e a c a c d K n K n . . . b d e c a e b d g b d c b b e . . . . . . . . . a c a c a d g g g f d f d f e e b Folie 12 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  7. Fakultät für Mathematik und Informatik Sketch of proof technique: (i) = ⇒ (ii): Prove: Bob wins on any forbidden configuration F j . Folie 13 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  8. Fakultät für Mathematik und Informatik Sketch of proof technique: (i) = ⇒ (ii): Prove: Bob wins on any forbidden configuration F j . (ii) = ⇒ (iii): Structural characterisation of graphs not containing any F j (this is the hard part of the proofs!) Folie 13 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  9. Fakultät für Mathematik und Informatik Sketch of proof technique: (i) = ⇒ (ii): Prove: Bob wins on any forbidden configuration F j . (ii) = ⇒ (iii): Structural characterisation of graphs not containing any F j (this is the hard part of the proofs!) (iii) = ⇒ (i): Prove: 1. Alice wins on any allowed structure E i 2. Every substructure of an allowed structure E i is again an allowed structure E i 0 Folie 13 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  10. Fakultät für Mathematik und Informatik Example for (i) = ⇒ (ii): Bob wins on “double fan” Folie 14 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  11. Fakultät für Mathematik und Informatik Example for (i) = ⇒ (ii): Bob wins on “double fan” Folie 14 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  12. Fakultät für Mathematik und Informatik Example for (i) = ⇒ (ii): Bob wins on “double fan” Folie 14 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  13. Fakultät für Mathematik und Informatik Example for (i) = ⇒ (ii): Bob wins on “double fan” Folie 14 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  14. Fakultät für Mathematik und Informatik Example for (i) = ⇒ (ii): Bob wins on “double fan” Folie 14 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  15. Fakultät für Mathematik und Informatik Example for (i) = ⇒ (ii): Bob wins on “double fan” Folie 14 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  16. Fakultät für Mathematik und Informatik Example for (iii) = ⇒ (i): Alice wins on “ear animal” H 0 K q K p K r H 4 H 1 H 3 H 2 Folie 15 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  17. Fakultät für Mathematik und Informatik Example for (iii) = ⇒ (i): Alice wins on “ear animal” H 0 K q K p K r H 4 H 1 H 3 H 2 Main goal is that the “ears” contain the same colours unless the second ear is fully coloured. And Alice should colour the central vertex as fast as possible. Folie 15 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  18. Fakultät für Mathematik und Informatik Idea of proof of (ii) = ⇒ (iii): Structural characterisation Structure of (nontrivial) Structure of (connected) [ B , B ] -perfect graphs [ B , − ] -perfect graphs (A. 2012): (with Edwin Lock 2016+): Folie 16 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  19. Fakultät für Mathematik und Informatik Idea of proof of (ii) = ⇒ (iii): Structural characterisation Structure of (nontrivial) Structure of (connected) [ B , B ] -perfect graphs [ B , − ] -perfect graphs (A. 2012): (with Edwin Lock 2016+): → dominating vertex exists − (by Lemma of Wolk (1965)) Folie 16 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  20. Fakultät für Mathematik und Informatik Idea of proof of (ii) = ⇒ (iii): Structural characterisation Structure of (nontrivial) Structure of (connected) [ B , B ] -perfect graphs [ B , − ] -perfect graphs (A. 2012): (with Edwin Lock 2016+): → dominating vertex exists − (by Lemma of Wolk (1965)) H 0 K q K p K r H 4 H 1 H 3 H 2 Folie 16 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  21. Fakultät für Mathematik und Informatik Idea of proof of (ii) = ⇒ (iii): Structural characterisation Structure of (nontrivial) Structure of (connected) [ B , B ] -perfect graphs [ B , − ] -perfect graphs (A. 2012): (with Edwin Lock 2016+): → dominating vertex exists − (by Lemma of Wolk (1965)) H 0 K q K p K r H 4 H 1 H 3 H 2 inner structure simple (1 page of case distinctions) Folie 16 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  22. Fakultät für Mathematik und Informatik Idea of proof of (ii) = ⇒ (iii): Structural characterisation Structure of (nontrivial) Structure of (connected) [ B , B ] -perfect graphs [ B , − ] -perfect graphs (A. 2012): (with Edwin Lock 2016+): → dominating vertex exists → dominating edge exists − − (by Lemma of Wolk (1965)) (Cozzens&Kelleher (1990)) H 0 K q K p K r H 4 H 1 H 3 H 2 inner structure simple (1 page of case distinctions) Folie 16 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  23. Fakultät für Mathematik und Informatik Idea of proof of (ii) = ⇒ (iii): Structural characterisation Structure of (nontrivial) Structure of (connected) [ B , B ] -perfect graphs [ B , − ] -perfect graphs (A. 2012): (with Edwin Lock 2016+): → dominating vertex exists → dominating edge exists − − (by Lemma of Wolk (1965)) (Cozzens&Kelleher (1990)) H 0 K q K p K r H 4 H 1 H 3 H 2 inner structure simple (1 page of case distinctions) Folie 16 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  24. Fakultät für Mathematik und Informatik Idea of proof of (ii) = ⇒ (iii): Structural characterisation Structure of (nontrivial) Structure of (connected) [ B , B ] -perfect graphs [ B , − ] -perfect graphs (A. 2012): (with Edwin Lock 2016+): → dominating vertex exists → dominating edge exists − − (by Lemma of Wolk (1965)) (Cozzens&Kelleher (1990)) H 0 K q K p K r H 4 H 1 —inner structure simple H 3 H 2 —examine structure of adjacencies bet- ween the three parts inner structure simple (20 pages of case distinctions) (1 page of case distinctions) Folie 16 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  25. Fakultät für Mathematik und Informatik Forbidden induced subgraphs for game-perfectness [ B , A ] [ A , A ] ? ? b c d c b c e e e a b c d a a e a b d d a b c a b g b e c d e a b c d e d e f c d a f f b d a a c f g c e c a g e b d e c a e b d g b d c b b e a g c a g c a d g f d f d f e e b Lock (2016) [ B , − ] A. (2012) [ B , B ] A. (2012) [ A , − ] , [ A , B ] Folie 17 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  26. Fakultät für Mathematik und Informatik 2. Digraphs Folie 18 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  27. Fakultät für Mathematik und Informatik Digraphs digraph D Folie 19 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  28. Fakultät für Mathematik und Informatik Digraphs symmetric part S ( D ) digraph D Folie 19 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  29. Fakultät für Mathematik und Informatik Digraphs symmetric part S ( D ) digraph D oriented part O ( D ) Folie 19 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  30. Fakultät für Mathematik und Informatik Dichromatic number, symmetric cliques, perfect digraphs The dichromatic number χ ( D ) of a digraph D is the smallest number of induced acyclic subdigraphs of D that cover the vertices of D . [Neumann-Lara 1982] Folie 20 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  31. Fakultät für Mathematik und Informatik Dichromatic number, symmetric cliques, perfect digraphs The dichromatic number χ ( D ) of a digraph D is the smallest number of induced acyclic subdigraphs of D that cover the vertices of D . [Neumann-Lara 1982] − → no monochromatic directed cycles! Folie 20 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  32. Fakultät für Mathematik und Informatik Dichromatic number, symmetric cliques, perfect digraphs The dichromatic number χ ( D ) of a digraph D is the smallest number of induced acyclic subdigraphs of D that cover the vertices of D . [Neumann-Lara 1982] − → no monochromatic directed cycles! A symmetric clique is a complete digraph (without loops) identical to its symmetric part. ω ( D ) = size of largest symmetric clique. Folie 20 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  33. Fakultät für Mathematik und Informatik Dichromatic number, symmetric cliques, perfect digraphs The dichromatic number χ ( D ) of a digraph D is the smallest number of induced acyclic subdigraphs of D that cover the vertices of D . [Neumann-Lara 1982] − → no monochromatic directed cycles! A symmetric clique is a complete digraph (without loops) identical to its symmetric part. ω ( D ) = size of largest symmetric clique. A digraph D is perfect if, for any induced subdigraph H of D , ω ( H ) = χ ( H ) . Folie 20 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  34. Fakultät für Mathematik und Informatik A generalization of the Strong Perfect Graph Theorem Theorem (A.&Hochstättler(2015)) A digraph is perfect if and only if it does not contain induced subdigraphs of the following types: (1) filled odd holes: i.e. D with S ( D ) is odd hole resp. (2) filled odd antiholes: i.e. D with S ( D ) is odd antihole resp. (3) directed cycles of length ≥ 3 . type (1) type (2) type (3) Folie 21 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  35. Fakultät für Mathematik und Informatik Perfectness is non-closed by taking complements perfect non perfect Folie 22 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  36. Fakultät für Mathematik und Informatik 2. a) Weakly game-perfect digraphs Folie 23 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  37. Fakultät für Mathematik und Informatik Weak digraph colouring games [Yang&Zhu(2010)] Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , such that they do not create monochromatic directed cycles. Alice wins if every vertex is coloured. Folie 24 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  38. Fakultät für Mathematik und Informatik Weak digraph colouring games [Yang&Zhu(2010)] Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , such that they do not create monochromatic directed cycles. Alice wins if every vertex is coloured. Folie 24 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  39. Fakultät für Mathematik und Informatik Weak digraph colouring games [Yang&Zhu(2010)] Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , such that they do not create monochromatic directed cycles. Alice wins if every vertex is coloured. Folie 24 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  40. Fakultät für Mathematik und Informatik Weak digraph colouring games [Yang&Zhu(2010)] Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , such that they do not create monochromatic directed cycles. Alice wins if every vertex is coloured. Folie 24 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  41. Fakultät für Mathematik und Informatik Weak digraph colouring games [Yang&Zhu(2010)] Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , such that they do not create monochromatic directed cycles. Alice wins if every vertex is coloured. Folie 24 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  42. Fakultät für Mathematik und Informatik Weak digraph colouring games [Yang&Zhu(2010)] Alice wins! Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , such that they do not create monochromatic directed cycles. Alice wins if every vertex is coloured. Folie 24 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  43. Fakultät für Mathematik und Informatik Weakly game-perfect digraphs Definition (weak game chromatic number) The weak game chromatic number χ wg ( D ) of a digraph D is the smallest number of colours so that Alice has a winning strategy for the weak colouring game played on D . Definition (weakly game-perfect digraph) A digraph D is weakly game-perfect (or wg -perfect) if, for any induced subdigraph H of D , ω ( H ) = χ wg ( H ) . Folie 25 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  44. Fakultät für Mathematik und Informatik Characterisation of weakly game-perfect digraphs Lemma If D does not contain an induced cycle � C n , n ≥ 3 , then every directed cycle has a (symmetric) edge as a chord. Theorem For a weak game wg and the associated undirected game g, a digraph D is wg-perfect if and only if 1. its symmetric part S ( D ) is a g-perfect graph and 2. D does not contain any directed cycle � C n with n ≥ 3 as an induced subdigraph. Folie 26 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  45. Fakultät für Mathematik und Informatik Characterisation of weakly game-perfect digraphs Lemma If D does not contain an induced cycle � C n , n ≥ 3 , then every directed cycle has a (symmetric) edge as a chord. Theorem For a weak game wg and the associated undirected game g, a digraph D is wg-perfect if and only if 1. its symmetric part S ( D ) is a g-perfect graph and 2. D does not contain any directed cycle � C n with n ≥ 3 as an induced subdigraph. If 2. is not true: D is not perfect, thus D is not wg -perfect. Folie 26 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  46. Fakultät für Mathematik und Informatik Characterisation of weakly game-perfect digraphs Lemma If D does not contain an induced cycle � C n , n ≥ 3 , then every directed cycle has a (symmetric) edge as a chord. Theorem For a weak game wg and the associated undirected game g, a digraph D is wg-perfect if and only if 1. its symmetric part S ( D ) is a g-perfect graph and 2. D does not contain any directed cycle � C n with n ≥ 3 as an induced subdigraph. If 2. is true but 1. is not true, i.e. S ( D ) is non- g -perfect. ⇒ ∃ subdigraph H , so that Bob wins on S ( H ) . He uses the same strategy for the play on H . Whenever he would close a monochromatic directed cycle, by Lemma this cycle would be monochromatic edge or have a monochromatic edge as a chord, contradicting the fact that he has a strategy for S ( H ) . ⇒ D is not wg -perfect. Folie 26 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  47. Fakultät für Mathematik und Informatik Characterisation of weakly game-perfect digraphs Lemma If D does not contain an induced cycle � C n , n ≥ 3 , then every directed cycle has a (symmetric) edge as a chord. Theorem For a weak game wg and the associated undirected game g, a digraph D is wg-perfect if and only if 1. its symmetric part S ( D ) is a g-perfect graph and 2. D does not contain any directed cycle � C n with n ≥ 3 as an induced subdigraph. If 1. and 2. are true: Let H be induced subdigraph of D . ⇒ Alice has winning strategy on S ( H ) . She uses the same strategy for the play on H . Whenever she would close a monochromatic directed cycle, by Lemma this cycle would be monochromatic edge or have a monochromatic edge as a chord, contradicting the fact that she has a strategy for S ( H ) . ⇒ D is wg -perfect. Folie 26 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  48. Fakultät für Mathematik und Informatik 2. b) Strongly game-perfect digraphs Folie 27 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  49. Fakultät für Mathematik und Informatik Strong digraph colouring games Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , which is different from colours of its in -neighbours. Alice wins if every vertex is coloured. Bob wins if a vertex is surrounded by all colours. Folie 28 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  50. Fakultät für Mathematik und Informatik Strong digraph colouring games Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , which is different from colours of its in -neighbours. Alice wins if every vertex is coloured. Bob wins if a vertex is surrounded by all colours. Folie 28 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  51. Fakultät für Mathematik und Informatik Strong digraph colouring games Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , which is different from colours of its in -neighbours. Alice wins if every vertex is coloured. Bob wins if a vertex is surrounded by all colours. Folie 28 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  52. Fakultät für Mathematik und Informatik Strong digraph colouring games Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , which is different from colours of its in -neighbours. Alice wins if every vertex is coloured. Bob wins if a vertex is surrounded by all colours. Folie 28 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  53. Fakultät für Mathematik und Informatik Strong digraph colouring games Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , which is different from colours of its in -neighbours. Alice wins if every vertex is coloured. Bob wins if a vertex is surrounded by all colours. Folie 28 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  54. Fakultät für Mathematik und Informatik Strong digraph colouring games Bob wins! Alice and Bob alternately colour uncoloured vertices of D with a colour from the set C , which is different from colours of its in -neighbours. Alice wins if every vertex is coloured. Bob wins if a vertex is surrounded by all colours. Folie 28 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  55. Fakultät für Mathematik und Informatik Strongly game-perfect digraphs Definition (game chromatic number) The (strong) game chromatic number χ g ( D ) of a digraph D is the smallest number of colours so that Alice has a winning strategy for the strong colouring game played on D . Definition (strongly game-perfect digraph) A digraph D is (strongly) game-perfect (or g -perfect) if, for any induced subdigraph H of D , ω ( H ) = χ g ( H ) . [6 types of games] Folie 29 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  56. Fakultät für Mathematik und Informatik Trivial if Bob begins: No additional structures Folie 30 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  57. Fakultät für Mathematik und Informatik Trivial if Bob begins: No additional structures Folie 30 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  58. Fakultät für Mathematik und Informatik Trivial if Bob begins: No additional structures A single arc is not [ B , ∗ ] -perfect. Folie 30 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  59. Fakultät für Mathematik und Informatik Trivial if Bob begins: No additional structures A single arc is not [ B , ∗ ] -perfect. Theorem For the games, where Bob begins, the class of game-perfect digraphs is equal to its subclass the class of game-perfect undirected graphs. Folie 30 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  60. Fakultät für Mathematik und Informatik Nontrivial if Alice begins: game [ A , A ] on paths The 48 [ A , A ] -perfect paths All minimal forbidden paths (red: non [ A , − ] /non [ A , B ] -perfect) for the game [ A , A ] a b c a b a b a b c a b c a b c a b c a b c d Q 0 = P 1 Q 1 Q 2 = P 2 Q 3 Q 4 Q 5 Q 6 = P 3 Q 7 F 3 , 1 a b c d a b c d a b c d a b c d a b c d a b c Q 8 Q 9 Q 10 Q 11 Q 12 a b c d a b c d a b c d e a b c d e a b c d e F 3 , 2 Q 13 Q 14 = P 4 Q 15 Q 16 Q 17 a b c d a b c d e a b c d e a b c d e a b c d e F 4 Q 18 Q 19 Q 20 Q 21 a b c d e a b c d e a b c d e a b c d e f a b c d e f Q 22 Q 23 Q 24 Q 25 F 5 , 1 a b c d e f a b c d e f a b c d e f a b c d e Q 26 Q 27 Q 28 a b c d e f a b c d e f a b c d e f F 5 , 2 Q 29 Q 30 Q 31 a b c d e f g a b c d e f a b c d e f a b c d e f F 7 , 1 Q 32 Q 33 Q 34 a b c d e f g a b c d e f g a b c d e f g a b c d e f g Q 35 Q 36 Q 37 F 7 , 2 a b c d e f g a b c d e f g a b c d e f g a b c d e f g h Q 38 Q 39 Q 40 a b c d e f g a b c d e f g h a b c d e f g h F 8 Q 41 Q 42 Q 43 a b c d e f g h a b c d e f g h Q 44 Q 45 a b c d e f g h i a b c d e f g h i Q 46 Q 47 Folie 31 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  61. Fakultät für Mathematik und Informatik ...if Alice begins: game [ A , A ] on cycles The 14 [ A , A ] -perfect cycles Forbidden cycles (red: non [ A , − ] /non [ A , B ] -perfect) for the game [ A , A ] b b b b b b b a a a a a a a c c c c c c c O 9 O 10 O 11 O 1 O 2 O 3 O 4 b c b c b c a a a d d d e e e O 12 O 13 O 14 O 15 O 16 O 5 O 6 O 7 b c b c b c b c b c a a a a d d d d e e e e a d O 17 O 18 O 19 O 20 f e b c b c O 8 a a d d Observation f e f e O 21 O 22 Let C be a cycle with n ≥ 7 vertices. Then C is not game-perfect. Folie 32 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  62. Fakultät für Mathematik und Informatik Kernels in digraphs A kernel S of digraph D = ( V , A ) : S V \ S S independent and absorbing Folie 33 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  63. Fakultät für Mathematik und Informatik Kernels in digraphs A kernel S of digraph D = ( V , A ) : Theorem (A.&Hochstättler(2015)) The complement of a perfect digraph has a kernel. S V \ S S independent and absorbing Folie 33 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  64. Fakultät für Mathematik und Informatik Kernels in digraphs A kernel S of digraph D = ( V , A ) : Theorem (A.&Hochstättler(2015)) The complement of a perfect digraph has a kernel. Theorem [ A , − ] -perfect digraphs are the complement of a perfect digraph. S V \ S S independent and absorbing Folie 33 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  65. Fakultät für Mathematik und Informatik Kernels in digraphs A kernel S of digraph D = ( V , A ) : Theorem (A.&Hochstättler(2015)) The complement of a perfect digraph has a kernel. Theorem [ A , − ] -perfect digraphs are the complement of a perfect digraph. S V \ S Corollary S independent and [ A , − ] -perfect digraphs have a kernel. absorbing Folie 33 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  66. Fakultät für Mathematik und Informatik Kernels in [ A , A ] -perfect digraphs We can show similarly Theorem Every [ A , A ] -perfect digraph D that does not contain the complement of a directed cycle � C 4 has a kernel. (a) (b) [ A , A ] -perfect digraphs: (a) does not have a kernel (b) has a kernel Open Question Characterise the [ A , A ] -perfect digraphs that contain a � C C 4 : which of them have a kernel? Folie 34 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  67. Fakultät für Mathematik und Informatik The remaining undirected cases [ B , A ] - resp. [ A , A ] -perfect graphs cannot be described by a finite list of minimal forbidden configurations, since every odd antihole is a minimal forbidden configuration. Open Question Characterize [ B , A ] - and [ A , A ] -perfect undirected graphs. Folie 35 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  68. Fakultät für Mathematik und Informatik On characterizing game-perfect digraphs For the games, where Alice begins, we have the following open problems. Problem Characterize (strongly) game-perfect superorientations of trees. Folie 36 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  69. Fakultät für Mathematik und Informatik On characterizing game-perfect digraphs For the games, where Alice begins, we have the following open problems. Problem Characterize (strongly) game-perfect superorientations of trees. Problem Characterize (strongly) game-perfect digraphs by a complete list of forbidden configurations. Folie 36 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  70. Fakultät für Mathematik und Informatik Complexity Open Question What is the complexity of the (di)graph colouring games? Folie 37 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  71. Fakultät für Mathematik und Informatik Thank you Thank you for your attention. Folie 38 GAGW 2017 Diskrete Mathematik und Optimierung Dominique Andres Edwin Lock

  72. ❖♣❡♥ q✉❡st✐♦♥s ♦♥ ❣❛♠❡✲♣❡r❢❡❝t ❣r❛♣❤s ❛♥❞ ❞✐❣r❛♣❤s ❖♣❡♥ ◗✉❡st✐♦♥s ♦♥ ●❛♠❡✲♣❡r❢❡❝t ●r❛♣❤s ❛♥❞ ❉✐❣r❛♣❤s ∗ ❙t❡♣❤❛♥ ❉♦♠✐♥✐q✉❡ ❆♥❞r❡s ❞♦♠✐♥✐q✉❡✳❛♥❞r❡s❅❢❡r♥✉♥✐✲❤❛❣❡♥✳❞❡ ●❛♠❡s ❛♥❞ ●r❛♣❤s ❲♦r❦s❤♦♣✱ ✷✸✕✷✺ ❖❝t♦❜❡r ✷✵✶✼ ✶ ✴ ✺

  73. ❖♣❡♥ q✉❡st✐♦♥s ♦♥ ❣❛♠❡✲♣❡r❢❡❝t ❣r❛♣❤s ❛♥❞ ❞✐❣r❛♣❤s ❈♦♥t❡♥ts ❋✐♥✐t❡♥❡ss ♦❢ t❤❡ s❡t ♦❢ ♠✐♥✐♠❛❧ ❢♦r❜✐❞❞❡♥ tr❡❡s ✶ ❊q✉✐✈❛❧❡♥❝❡ ♦❢ [ ❆ , − ] ✲ ❛♥❞ [ ❆ , ❇ ] ✲♣❡r❢❡❝t♥❡ss ✷ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ χ ✲❝♦❧✲♣❡r❢❡❝t ❣r❛♣❤s ✸ ✷ ✴ ✺

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