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Interval Graphs and (Normal Helly) Circular-arc Graphs Yixin Cao Department of Computing, Hong Kong Polytechnic University Constrained Recognition Problems (ICALP 2018) July 9,

  1. Circular-arc Helly circular-arc Normal circular-arc Proper circular-arc Normal Helly circular-arc Chordal W 4 , S 3 claw holes Proper Helly circular-arc Unit circular-arc Interval W 4 Unit Helly circular-arc claw holes [Lin et al. 2013]; [C 2017] Unit interval = Proper interval 28 / 1

  2. Circular-arc Helly circular-arc Normal circular-arc Proper circular-arc Normal Helly circular-arc Chordal W 4 , S 3 claw holes Proper Helly circular-arc Unit circular-arc Interval W 4 Unit Helly circular-arc claw holes [Lin et al. 2013]; [C 2017] Unit interval = Proper interval 29 / 1

  3. Circular-arc Helly circular-arc Normal circular-arc Proper circular-arc Normal Helly circular-arc Chordal W 4 , S 3 claw holes Proper Helly circular-arc Unit circular-arc Interval W 4 Unit Helly circular-arc claw holes [Lin et al. 2013]; [C 2017] Unit interval = Proper interval 30 / 1

  4. Unit interval graphs Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval holes claw Unit interval = Proper interval 31 / 1

  5. Unit interval graphs v 6 v 2 v 4 v 2 v 8 v 6 v 3 v 4 v 7 v 1 v 5 v 7 v 1 v 3 v 5 v 6 v 3 v 2 v 6 v 7 v 5 v 1 v 2 v 1 v 4 v 7 v 8 v 4 v 5 v 3 The left is a unit interval graph; the right is not. 32 / 1

  6. Unit interval graphs v 6 v 2 v 4 v 2 v 8 v 6 v 3 v 4 v 7 v 1 v 5 v 7 v 1 v 3 v 5 v 6 v 3 v 2 v 6 v 7 v 5 v 1 v 2 v 1 v 4 v 7 v 8 v 4 v 5 v 3 The left is a unit interval graph; the right is not. 33 / 1

  7. Forbidden induced subgraphs [Wegner 1967] · · · C 4 C 5 tent net claw unit interval ⊂ interval ⊂ chordal 34 / 1

  8. Forbidden induced subgraphs [Wegner 1967] · · · C 4 C 5 tent net claw unit interval ⊂ interval ⊂ chordal 35 / 1

  9. Unit interval vertex deletion Unit interval vertex deletion Input: A graph G and an integer k . Task: A set V − of ≤ k vertices such that G − V − is a unit interval graph. NP-complete [Lewis & Yannakakis 1978] O ((14 k + 14) k +1 · kn 6 ) O (6 k · n 6 ) O (6 k · ( n + m )) FPT [Marx 2006] [van Bevern et al. 2010] [Villanger 2013] [C 2017] 36 / 1

  10. Unit interval vertex deletion Unit interval vertex deletion Input: A graph G and an integer k . Task: A set V − of ≤ k vertices such that G − V − is a unit interval graph. NP-complete [Lewis & Yannakakis 1978] O ((14 k + 14) k +1 · kn 6 ) O (6 k · n 6 ) O (6 k · ( n + m )) FPT [Marx 2006] [van Bevern et al. 2010] [Villanger 2013] [C 2017] 37 / 1

  11. Main ideas Standard technique A small subgraph F can be found in n | F | time and dealt with an | F | -way branching. Make it { claw , net , tent } -free, then solve it using chordal vertex deletion [van Bevern et al. 2010] Make it { claw , net , tent , C 4 , C 5 , C 6 } -free, and then use iterative compression. [Villanger 2013] A connected { claw , net , tent , C 4 , C 5 , C 6 } -free graphs are proper circular-arc graphs, on which the problem can be solved in linear time. (by manually building a proper circular-arc model.) 38 / 1

  12. Proper Helly circular-arc graphs A graph having a circular-arc model that is both proper and Helly . a a c a b b b c A proper model A Helly model 39 / 1

  13. Proper Helly circular-arc graphs A graph having a circular-arc model that is both proper and Helly . a a c a b b b c A proper model A Helly model 40 / 1

  14. Why proper Helly? Theorem (Tucker 1974; Lin et al. 2013) A graph is a proper Helly circular-arc graph if and only if it contains no claw, net, tent, W 4 , W 5 , C 6 , or C ∗ ℓ for ℓ ≥ 4 (a hole C ℓ and another isolated vertex). A trivial corollary: If a proper Helly circular-arc graph is chordal, then it is a unit interval graph. A nontrivial corollary: A connected { claw , net , tent , C 4 , C 5 } -free graph is a proper Helly circular-arc graph. 41 / 1

  15. Why proper Helly? Theorem (Tucker 1974; Lin et al. 2013) A graph is a proper Helly circular-arc graph if and only if it contains no claw, net, tent, W 4 , W 5 , C 6 , or C ∗ ℓ for ℓ ≥ 4 (a hole C ℓ and another isolated vertex). A trivial corollary: If a proper Helly circular-arc graph is chordal, then it is a unit interval graph. A nontrivial corollary: A connected { claw , net , tent , C 4 , C 5 } -free graph is a proper Helly circular-arc graph. 42 / 1

  16. Why proper Helly? Theorem (Tucker 1974; Lin et al. 2013) A graph is a proper Helly circular-arc graph if and only if it contains no claw, net, tent, W 4 , W 5 , C 6 , or C ∗ ℓ for ℓ ≥ 4 (a hole C ℓ and another isolated vertex). A trivial corollary: If a proper Helly circular-arc graph is chordal, then it is a unit interval graph. A nontrivial corollary: A connected { claw , net , tent , C 4 , C 5 } -free graph is a proper Helly circular-arc graph. 43 / 1

  17. Achilles’ heel α Once all claws, nets, tents, C 4 ’s, and C 5 ’s destroyed, it suffices to find the thinnest point from the model. 44 / 1

  18. Break time You may safely skip the following three slides if you are tired. 45 / 1

  19. How about unit Helly circular-arc graphs? Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval Unit Helly circular-arc claw holes This is actually the CI ( ℓ, 1) graph defined Unit interval = Proper interval by [Tucker 1974]; see also [Lin et al. 2013]. = proper Helly circular-arc ∩ chordal = unit Helly circular-arc ∩ chordal 46 / 1

  20. How about unit Helly circular-arc graphs? Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval Unit Helly circular-arc claw holes This is actually the CI ( ℓ, 1) graph defined Unit interval = Proper interval by [Tucker 1974]; see also [Lin et al. 2013]. = proper Helly circular-arc ∩ chordal = unit Helly circular-arc ∩ chordal 47 / 1

  21. How about unit Helly circular-arc graphs? Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval Unit Helly circular-arc claw holes This is actually the CI ( ℓ, 1) graph defined Unit interval = Proper interval by [Tucker 1974]; see also [Lin et al. 2013]. = proper Helly circular-arc ∩ chordal = unit Helly circular-arc ∩ chordal 48 / 1

  22. How about unit Helly circular-arc graphs? Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval Unit Helly circular-arc claw holes This is actually the CI ( ℓ, 1) graph defined Unit interval = Proper interval by [Tucker 1974]; see also [Lin et al. 2013]. = proper Helly circular-arc ∩ chordal = unit Helly circular-arc ∩ chordal 49 / 1

  23. Edge deletion proper Helly circular-arc → unit interval by deleting edges: Achilles’ heel with respect to edges. The thinnest point for vertices is α α The thinnest point for edges is β β 50 / 1

  24. A slightly stronger statement [van Bevern et al. 2010] Unit interval vertex deletion remains NP-hard on { claw , net , tent } -free graphs. [Villanger 2013] Unit interval vertex deletion is in P for { claw , net , tent , C 4 , C 5 , C 6 } -free graph. 7 [Villanger 2013] How about { claw , net , tent , C 4 } -free graphs? 7 20 10 10 [C 2017] If a connected { claw , net , tent , C 4 } -free graph is not 21 4 a proper Helly circular-arc graph, then it is a fat W 5 . 51 / 1

  25. A slightly stronger statement [van Bevern et al. 2010] Unit interval vertex deletion remains NP-hard on { claw , net , tent } -free graphs. [Villanger 2013] Unit interval vertex deletion is in P for { claw , net , tent , C 4 , C 5 , C 6 } -free graph. 7 [Villanger 2013] How about { claw , net , tent , C 4 } -free graphs? 7 20 10 10 [C 2017] If a connected { claw , net , tent , C 4 } -free graph is not 21 4 a proper Helly circular-arc graph, then it is a fat W 5 . 52 / 1

  26. A slightly stronger statement [van Bevern et al. 2010] Unit interval vertex deletion remains NP-hard on { claw , net , tent } -free graphs. [Villanger 2013] Unit interval vertex deletion is in P for { claw , net , tent , C 4 , C 5 , C 6 } -free graph. 7 [Villanger 2013] How about { claw , net , tent , C 4 } -free graphs? 7 20 10 10 [C 2017] If a connected { claw , net , tent , C 4 } -free graph is not 21 4 a proper Helly circular-arc graph, then it is a fat W 5 . 53 / 1

  27. A slightly stronger statement [van Bevern et al. 2010] Unit interval vertex deletion remains NP-hard on { claw , net , tent } -free graphs. [Villanger 2013] Unit interval vertex deletion is in P for { claw , net , tent , C 4 , C 5 , C 6 } -free graph. 7 [Villanger 2013] How about { claw , net , tent , C 4 } -free graphs? 7 20 10 10 [C 2017] If a connected { claw , net , tent , C 4 } -free graph is not 21 4 a proper Helly circular-arc graph, then it is a fat W 5 . 54 / 1

  28. Normal Helly circular-arc graphs Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval holes claw Unit interval = Proper interval 55 / 1

  29. The problems Characterization (by forbidden induced subgraphs): Identify the set H of minimal subgraphs such that G is a normal Helly circular-arc graphs if and only if it contains no subgraph in H . Recognition: Efficiently decide whether a given graph is a normal Helly circular-arc graph or not. Detection: Either a model that is both normal and Helly (positive certificate), or a forbidden induced subgraph (negative certificate). 56 / 1

  30. The problems Characterization (by forbidden induced subgraphs): Identify the set H of minimal subgraphs such that G is a normal Helly circular-arc graphs if and only if it contains no subgraph in H . Recognition: Efficiently decide whether a given graph is a normal Helly circular-arc graph or not. Detection: Either a model that is both normal and Helly (positive certificate), or a forbidden induced subgraph (negative certificate). 57 / 1

  31. The problems Characterization (by forbidden induced subgraphs): Identify the set H of minimal subgraphs such that G is a normal Helly circular-arc graphs if and only if it contains no subgraph in H . Recognition: Efficiently decide whether a given graph is a normal Helly circular-arc graph or not. Detection: Either a model that is both normal and Helly (positive certificate), or a forbidden induced subgraph (negative certificate). 58 / 1

  32. Characterization of interval graphs Asteroidal triple (AT): Three vertices of which each pair is connected by a path avoiding neighbors of the third one. Theorem (Lekkerkerker and Boland, 1962) A graph is an interval graph if and only if it contains no holes or ATs. any hole of length ≥ 6 contains ATs. 59 / 1

  33. Characterization of interval graphs Asteroidal triple (AT): Three vertices of which each pair is connected by a path avoiding neighbors of the third one. Theorem (Lekkerkerker and Boland, 1962) A graph is an interval graph if and only if it contains no holes or ATs. any hole of length ≥ 6 contains ATs. 60 / 1

  34. Chordal asteroidal witnesses ( CAW ) Asteroidal witness: a minimal graph that contains an AT. All chordal asteroidal witnesses are minimal forbidden induced subgraphs of NHCAG. (Recall that normal Helly circular-arc ∩ chordal = interval .) We are henceforth focused on the non-chordal case, hence holes. 61 / 1

  35. Intuition In a normal Helly circular-arc model, Any minimal set of arcs covering the circle induces a hole. For any vertex v in a hole, 62 / 1

  36. Intuition In a normal Helly circular-arc model, v Any minimal set of arcs covering the circle induces a hole. For any vertex v in a hole, 63 / 1

  37. Intuition In a normal Helly circular-arc model, v Any minimal set of arcs covering the circle induces a hole. For any vertex v in a hole, G − N [ v ] is an interval subgraph. 64 / 1

  38. The auxiliary graph ℧ ( G ) Construction of ℧ ( G ) : ⋆ 1 find a vertex v with the largest degree; ( G − N [ v ] is an interval graph.) 2 append a copy of N [ v ] to “each end” of G − N [ v ] . 3 add a new vertex w to keep the left end of the left copy of N [ v ] . ⋆ : Upon a failure during this construction, a forbidden induced subgraph can be detected. Theorem (C 2016; C Grippo Safe 2017) G is a normal Helly circular-arc graph if and only if ℧ ( G ) is an interval graph. 65 / 1

  39. The auxiliary graph ℧ ( G ) Construction of ℧ ( G ) : ⋆ 1 find a vertex v with the largest degree; ( G − N [ v ] is an interval graph.) 2 append a copy of N [ v ] to “each end” of G − N [ v ] . 3 add a new vertex w to keep the left end of the left copy of N [ v ] . ⋆ : Upon a failure during this construction, a forbidden induced subgraph can be detected. Theorem (C 2016; C Grippo Safe 2017) G is a normal Helly circular-arc graph if and only if ℧ ( G ) is an interval graph. 66 / 1

  40. The auxiliary graph ℧ ( G ) Construction of ℧ ( G ) : ⋆ 1 find a vertex v with the largest degree; ( G − N [ v ] is an interval graph.) 2 append a copy of N [ v ] to “each end” of G − N [ v ] . 3 add a new vertex w to keep the left end of the left copy of N [ v ] . ⋆ : Upon a failure during this construction, a forbidden induced subgraph can be detected. Theorem (C 2016; C Grippo Safe 2017) G is a normal Helly circular-arc graph if and only if ℧ ( G ) is an interval graph. 67 / 1

  41. The auxiliary graph ℧ ( G ) Construction of ℧ ( G ) : ⋆ 1 find a vertex v with the largest degree; ( G − N [ v ] is an interval graph.) 2 append a copy of N [ v ] to “each end” of G − N [ v ] . 3 add a new vertex w to keep the left end of the left copy of N [ v ] . ⋆ : Upon a failure during this construction, a forbidden induced subgraph can be detected. Theorem (C 2016; C Grippo Safe 2017) G is a normal Helly circular-arc graph if and only if ℧ ( G ) is an interval graph. 68 / 1

  42. The auxiliary graph ℧ ( G ) Construction of ℧ ( G ) : ⋆ 1 find a vertex v with the largest degree; ( G − N [ v ] is an interval graph.) 2 append a copy of N [ v ] to “each end” of G − N [ v ] . 3 add a new vertex w to keep the left end of the left copy of N [ v ] . ⋆ : Upon a failure during this construction, a forbidden induced subgraph can be detected. Theorem (C 2016; C Grippo Safe 2017) G is a normal Helly circular-arc graph if and only if ℧ ( G ) is an interval graph. 69 / 1

  43. Circular-arc model for G ⇒ Interval model for ℧ ( G ) v 2 v 1 h − 1 1(0) h 0 every point in the model h − 2 has a value in (0 , 1] . 0 . 75 0 . 25 h 1 0 . 50 h 2 0 a 1 1 + a v r v l 2 2 h r w h l 0 0 h r h l v r h r v l h l − 1 − 1 1 1 1 1 L R 70 / 1

  44. Circular-arc model for G ⇒ Interval model for ℧ ( G ) v 2 v 1 h − 1 1(0) h 0 every point in the model h − 2 has a value in (0 , 1] . 0 . 75 0 . 25 h 1 0 . 50 h 2 0 a 1 1 + a v r v l 2 2 h r w h l 0 0 h r h l v r h r v l h l − 1 − 1 1 1 1 1 L R 71 / 1

  45. Other forbidden induced subgraphs (with holes) K 2 , 3 twin- C 5 C 6 domino C ∗ FIS-1 F wheel 72 / 1

  46. The certifying recognition algorithm 1. if G is chordal then return an interval model of G or a caw ; 2. build the auxiliary graph ℧ ( G ) ; 3. if ℧ ( G ) is an interval graph then build a normal and Helly circular-arc model A for G ; return A ; 4. else find a minimal forbidden induced subgraph F of G ; return F . 73 / 1

  47. Related subclasses of circular-arc graphs Characterization Certifying recognition Unknown † circular arc (ca) Unknown Unknown ‡ normal ca Unknown proper ca Tucker 1974 Kaplan&Nussbaum 2009 unit ca Tucker 1974 Kaplan&Nussbaum 2009 unit Helly ca Lin et al. 2013 Lin et al. 2013 proper Helly ca Lin et al. 2013 Lin et al. 2013 normal Helly ca C Grippo & Safe 2017 † : linear recognition is known. ‡ : circular arc graphs that are not normal are known. 74 / 1

  48. Interval graphs Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval holes claw Unit interval = Proper interval 75 / 1

  49. Characterization of interval graphs Asteroidal triple (AT): Hole: Three vertices of which each pair is connected an induced cycle of length ≥ 4 . by a path avoiding neighbors of the third one. normal Helly circular-arc ∩ chordal = interval . 76 / 1

  50. Characterization of interval graphs Asteroidal triple (AT): Hole: Three vertices of which each pair is connected an induced cycle of length ≥ 4 . by a path avoiding neighbors of the third one. normal Helly circular-arc ∩ chordal = interval . 77 / 1

  51. Reduction: small forbidden subgraphs Recall that Standard technique A small subgraph F can be found in n | F | time and dealt with an | F | -way branching. Kill all forbidden subgraphs of ≤ 10 vertices: The resulting graph is called reduced . 78 / 1

  52. Reduction: small forbidden subgraphs Recall that Standard technique A small subgraph F can be found in n | F | time and dealt with an | F | -way branching. Kill all forbidden subgraphs of ≤ 10 vertices: The resulting graph is called reduced . 79 / 1

  53. Shallow terminals We are left with long holes (at least 11 vertices) and s s c 1 c 2 c l r l r b 0 b 1 b 2 b i b d − 1 b d b d +1 b 0 b 1 b 2 b i b d − 1 b d b d +1 Shallow terminal: of the unique asteroidal triple, one vertex s has a shorter distance to the other two ( l, r ). 80 / 1

  54. Main theorem In a reduced graph, shallow terminals form modules (set of vertices with the same neighborhood); and neighbors of each of the modules induces a clique. Or (in the parlance of modular decomposition): Each shallow terminal in the quotient graph of a reduced graph is simplicial. 81 / 1

  55. Main theorem In a reduced graph, shallow terminals form modules (set of vertices with the same neighborhood); and neighbors of each of the modules induces a clique. Or (in the parlance of modular decomposition): Each shallow terminal in the quotient graph of a reduced graph is simplicial. 82 / 1

  56. Maximal cliques Shallow terminals are not in any holes; the rest form a normal Helly circular-arc graph. n maximal cliques chordal graph: tree interval graph: path normal Helly circular-arc graph: cycle reduced graph: olive ring. 83 / 1

  57. Linear-time 84 / 1

  58. Almost interval graphs Theorem (Yannakakis 79, 81; Goldberg et al. 95) All modification problems to interval graphs are NP-complete. interval + ke , interval − ke , and interval + kv can be recognized in time n O ( k ) (polynomial for fixed k ) [trivial]. interval − ke can be recognized in time k 2 k · n 5 : [Heggernes et al. STOC’07]; and interval + kv can be recognized in time k 9 · n 9 [Cao & Marx SODA’14]. f ( k ) · n O (1) : Fixed-parameter tractable (FPT) Can interval + ke be recognized in FPT time as well? Can any of them be recognized in linear time? 85 / 1

  59. Almost interval graphs Theorem (Yannakakis 79, 81; Goldberg et al. 95) All modification problems to interval graphs are NP-complete. interval + ke , interval − ke , and interval + kv can be recognized in time n O ( k ) (polynomial for fixed k ) [trivial]. interval − ke can be recognized in time k 2 k · n 5 : [Heggernes et al. STOC’07]; and interval + kv can be recognized in time k 9 · n 9 [Cao & Marx SODA’14]. f ( k ) · n O (1) : Fixed-parameter tractable (FPT) Can interval + ke be recognized in FPT time as well? Can any of them be recognized in linear time? 86 / 1

  60. Prime graphs Definition M ⊆ V ( G ) is a module of G if they have the same neighborhood outside M : u, v ∈ M and x �∈ M , u ∼ x iff v ∼ x . A graph G is prime if a module of G is V ( G ) or consists of a single vertex. Observation details omitted It suffices to solve the problem on prime graphs. 87 / 1

  61. Prime graphs Definition M ⊆ V ( G ) is a module of G if they have the same neighborhood outside M : u, v ∈ M and x �∈ M , u ∼ x iff v ∼ x . A graph G is prime if a module of G is V ( G ) or consists of a single vertex. Observation details omitted It suffices to solve the problem on prime graphs. 88 / 1

  62. Conclusion We have used the connection as a black box to devise a 10 k · n O (1) -time algorithm for the interval vertex deletion problem. Using it as a white box, the runtime can be improved to O (10 k · ( n + m )) . With more careful use of modules, we can solve the interval completion and interval edge deletion problem as well. 89 / 1

  63. Conclusion We have used the connection as a black box to devise a 10 k · n O (1) -time algorithm for the interval vertex deletion problem. Using it as a white box, the runtime can be improved to O (10 k · ( n + m )) . With more careful use of modules, we can solve the interval completion and interval edge deletion problem as well. 90 / 1

  64. Conclusion We have used the connection as a black box to devise a 10 k · n O (1) -time algorithm for the interval vertex deletion problem. Using it as a white box, the runtime can be improved to O (10 k · ( n + m )) . With more careful use of modules, we can solve the interval completion and interval edge deletion problem as well. 91 / 1

  65. Conclusion We have used the connection as a black box to devise a 10 k · n O (1) -time algorithm for the interval vertex deletion problem. Using it as a white box, the runtime can be improved to O (10 k · ( n + m )) . With more careful use of modules, we can solve the interval completion and interval edge deletion problem as well. 92 / 1

  66. Conclusion We have used the connection as a black box to devise a 10 k · n O (1) -time algorithm for the interval vertex deletion problem. Using it as a white box, the runtime can be improved to O (10 k · ( n + m )) . With more careful use of modules, we can solve the interval completion and interval edge deletion problem as well. 93 / 1

  67. Epilogue Normal Helly circular-arc Chordal claw holes Proper Helly circular-arc Interval holes claw Unit interval = Proper interval 94 / 1

  68. Edge deletions Conjecture a minimal solution of edge deletion is “local” to some point in an arc model for G . v 3 v 2 u 3 u 2 v 4 u 4 u 1 v 1 u 5 u 6 v 5 v 6 95 / 1

  69. Edge deletions Conjecture a minimal solution of edge deletion is “local” to some point in an arc model for G . v 3 v 2 u 3 u 2 no! v 4 u 4 u 1 v 1 u 5 u 6 v 5 v 6 96 / 1

  70. Edge deletions Conjecture a minimal solution of edge deletion is “local” to some point in an arc model for G . v 3 v 2 u 3 u 2 no! v 4 u 4 u 1 v 1 u 5 u 6 v 5 v 6 97 / 1

  71. Edge deletions Conjecture a minimal solution of edge deletion is “local” to some point in an arc model for G . v 3 v 2 u 3 u 2 no! v 4 u 4 u 1 v 1 u 5 u 6 v 5 v 6 98 / 1

  72. To break long holes Definition − → E ( α ) = { vu : α ∈ A v , α �∈ A u , v → u } , where v → u means that arc A v intersects arc A u from the left. α 0 ℓ A trivial corollary For any point α , the subgraph G − − → E ( α ) is a unit interval graph. 99 / 1

  73. To break long holes Definition − → E ( α ) = { vu : α ∈ A v , α �∈ A u , v → u } , where v → u means that arc A v intersects arc A u from the left. α 0 ℓ A trivial corollary For any point α , the subgraph G − − → E ( α ) is a unit interval graph. 100 / 1

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