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On Separators in Temporal Graphs Hendrik Molter Algorithmics and Computational Complexity, TU Berlin, Germany Algorithmic Aspects of Temporal Graphs, Satellite Workshop of ICALP 2018, Prague Based on joint work with Till Fluschnik, Rolf

  1. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  2. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  3. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  4. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  5. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  6. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  7. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  8. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  9. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  10. Introduction Temporal Separators: Definition and Related Work (Non-)Strict ( s , z ) -Separation Input: A temporal graph G = ( V , E 1 ,..., E τ ) with two distinct vertices s , z ∈ V , and an integer k . Question: Is there a subset S ⊆ V \{ s , z } of size at most k such that there is no (non-)strict ( s , z ) -path in G − S ? Berman [1996, Networks] showed that for temporal graphs Menger’s Theorem fails (vertex-variant). 5 3 s z G : 1 2 6 7 4 The edge-deletion variant can be computed in polynomial-time. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 5 / 23

  11. Introduction Related Work II Kempe, Kleinberg, and Kumar [2002, JCSS] showed that (Non-)Strict ( s , z ) -Separation is NP-hard. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 6 / 23

  12. Introduction Related Work II Kempe, Kleinberg, and Kumar [2002, JCSS] showed that (Non-)Strict ( s , z ) -Separation is NP-hard. Menger’s Theorem holds if the underlying graph excludes a fixed minor. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 6 / 23

  13. Introduction Related Work II Kempe, Kleinberg, and Kumar [2002, JCSS] showed that (Non-)Strict ( s , z ) -Separation is NP-hard. Menger’s Theorem holds if the underlying graph excludes a fixed minor. s z Hendrik Molter, TU Berlin On Separators in Temporal Graphs 6 / 23

  14. Introduction Related Work II Kempe, Kleinberg, and Kumar [2002, JCSS] showed that (Non-)Strict ( s , z ) -Separation is NP-hard. Menger’s Theorem holds if the underlying graph excludes a fixed minor. s z This presentation is based on Fluschnik et al. [2018, WG] and Zschoche et al. [2018, MFCS]. (Both to appear, available on arXiv.) Hendrik Molter, TU Berlin On Separators in Temporal Graphs 6 / 23

  15. Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. n : instance size k : parameter Hendrik Molter, TU Berlin On Separators in Temporal Graphs 7 / 23

  16. Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. XP : Solvable in n g ( k ) time. n : instance size k : parameter Hendrik Molter, TU Berlin On Separators in Temporal Graphs 7 / 23

  17. Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. XP : Solvable in n g ( k ) time. Parameterized Hardness W[1]-hard : Presumably no FPT algorithm (XP algorithm possible). n : instance size k : parameter Hendrik Molter, TU Berlin On Separators in Temporal Graphs 7 / 23

  18. Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. XP : Solvable in n g ( k ) time. Parameterized Hardness W[1]-hard : Presumably no FPT algorithm (XP algorithm possible). para-NP-hard : NP-hard for constant k (no XP algorithm). n : instance size k : parameter Hendrik Molter, TU Berlin On Separators in Temporal Graphs 7 / 23

  19. Complexity of Finding Temporal Separators Basic Results Basic Results. ( s , z ) -Separation Parameter Strict Non-Strict 2 ≤ τ ≤ 4 poly-time para-NP-hard τ ≥ 5 para-NP-hard Hendrik Molter, TU Berlin On Separators in Temporal Graphs 8 / 23

  20. Complexity of Finding Temporal Separators Basic Results Basic Results. ( s , z ) -Separation Parameter Strict Non-Strict 2 ≤ τ ≤ 4 poly-time para-NP-hard τ ≥ 5 para-NP-hard k W[1]-hard W[1]-hard Hendrik Molter, TU Berlin On Separators in Temporal Graphs 8 / 23

  21. Complexity of Finding Temporal Separators Basic Results Basic Results. ( s , z ) -Separation Parameter Strict Non-Strict 2 ≤ τ ≤ 4 poly-time para-NP-hard τ ≥ 5 para-NP-hard k W[1]-hard W[1]-hard τ + k FPT open Hendrik Molter, TU Berlin On Separators in Temporal Graphs 8 / 23

  22. Complexity of Finding Temporal Separators Basic Results Basic Results. ( s , z ) -Separation Parameter Strict Non-Strict 2 ≤ τ ≤ 4 poly-time para-NP-hard τ ≥ 5 para-NP-hard k W[1]-hard W[1]-hard τ + k FPT open Hendrik Molter, TU Berlin On Separators in Temporal Graphs 8 / 23

  23. Complexity of Finding Temporal Separators Basic Results Basic Results. ( s , z ) -Separation Parameter Strict Non-Strict 2 ≤ τ ≤ 4 poly-time para-NP-hard τ ≥ 5 para-NP-hard k W[1]-hard W[1]-hard τ + k FPT open Canonical next step: Restrict input graphs. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 8 / 23

  24. Complexity of Finding Temporal Separators Basic Results Basic Results. ( s , z ) -Separation Parameter Strict Non-Strict 2 ≤ τ ≤ 4 poly-time para-NP-hard τ ≥ 5 para-NP-hard k W[1]-hard W[1]-hard τ + k FPT open Canonical next step: Restrict input graphs. Restrict each layer. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 8 / 23

  25. Complexity of Finding Temporal Separators Basic Results Basic Results. ( s , z ) -Separation Parameter Strict Non-Strict 2 ≤ τ ≤ 4 poly-time para-NP-hard τ ≥ 5 para-NP-hard k W[1]-hard W[1]-hard τ + k FPT open Canonical next step: Restrict input graphs. Restrict each layer. Restrict the underlying graph. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 8 / 23

  26. Complexity of Finding Temporal Separators Restricting each Layer (Non-)Strict ( s , z ) -Separation with restricted layers . Layer Restriction Complexity Hendrik Molter, TU Berlin On Separators in Temporal Graphs 9 / 23

  27. Complexity of Finding Temporal Separators Restricting each Layer (Non-)Strict ( s , z ) -Separation with restricted layers . Layer Restriction Complexity at most one edge NP-hard and W[1]-hard wrt. k Hendrik Molter, TU Berlin On Separators in Temporal Graphs 9 / 23

  28. Complexity of Finding Temporal Separators Restricting each Layer (Non-)Strict ( s , z ) -Separation with restricted layers . Layer Restriction Complexity at most one edge NP-hard and W[1]-hard wrt. k forest para-NP-hard wrt. τ unit interval Hendrik Molter, TU Berlin On Separators in Temporal Graphs 9 / 23

  29. Complexity of Finding Temporal Separators Restricting each Layer (Non-)Strict ( s , z ) -Separation with restricted layers . Layer Restriction Complexity at most one edge NP-hard and W[1]-hard wrt. k forest para-NP-hard wrt. τ unit interval Hendrik Molter, TU Berlin On Separators in Temporal Graphs 9 / 23

  30. Complexity of Finding Temporal Separators Restricting each Layer (Non-)Strict ( s , z ) -Separation with restricted layers . Layer Restriction Complexity at most one edge NP-hard and W[1]-hard wrt. k forest para-NP-hard wrt. τ unit interval Take away message: Layer restrictions do not help much. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 9 / 23

  31. Complexity of Finding Temporal Separators Restricting the Underlying Graph (Non-)Strict ( s , z ) -Separation with restricted underlying graph . Underlying Graph Restriction Complexity Hendrik Molter, TU Berlin On Separators in Temporal Graphs 10 / 23

  32. Complexity of Finding Temporal Separators Restricting the Underlying Graph (Non-)Strict ( s , z ) -Separation with restricted underlying graph . Underlying Graph Restriction Complexity poly-time (FPT wrt. tw + τ ) bounded treewidth Hendrik Molter, TU Berlin On Separators in Temporal Graphs 10 / 23

  33. Complexity of Finding Temporal Separators Restricting the Underlying Graph (Non-)Strict ( s , z ) -Separation with restricted underlying graph . Underlying Graph Restriction Complexity poly-time (FPT wrt. tw + τ ) bounded treewidth bounded vertex cover poly-time (FPT) Hendrik Molter, TU Berlin On Separators in Temporal Graphs 10 / 23

  34. Complexity of Finding Temporal Separators Restricting the Underlying Graph (Non-)Strict ( s , z ) -Separation with restricted underlying graph . Underlying Graph Restriction Complexity poly-time (FPT wrt. tw + τ ) bounded treewidth bounded vertex cover poly-time (FPT) complete − { s , z } bipartite para-NP-h wrt. τ / W[1]-h wrt. k line graph Hendrik Molter, TU Berlin On Separators in Temporal Graphs 10 / 23

  35. Complexity of Finding Temporal Separators Restricting the Underlying Graph (Non-)Strict ( s , z ) -Separation with restricted underlying graph . Underlying Graph Restriction Complexity poly-time (FPT wrt. tw + τ ) bounded treewidth bounded vertex cover poly-time (FPT) complete − { s , z } bipartite para-NP-h wrt. τ / W[1]-h wrt. k line graph NP-hard (Strict: FPT wrt. τ ) planar Hendrik Molter, TU Berlin On Separators in Temporal Graphs 10 / 23

  36. Complexity of Finding Temporal Separators Restricting the Underlying Graph (Non-)Strict ( s , z ) -Separation with restricted underlying graph . Underlying Graph Restriction Complexity poly-time (FPT wrt. tw + τ ) bounded treewidth bounded vertex cover poly-time (FPT) complete − { s , z } bipartite para-NP-h wrt. τ / W[1]-h wrt. k line graph NP-hard (Strict: FPT wrt. τ ) planar Hendrik Molter, TU Berlin On Separators in Temporal Graphs 10 / 23

  37. Complexity of Finding Temporal Separators Restricting the Underlying Graph (Non-)Strict ( s , z ) -Separation with restricted underlying graph . Underlying Graph Restriction Complexity poly-time (FPT wrt. tw + τ ) bounded treewidth bounded vertex cover poly-time (FPT) complete − { s , z } bipartite para-NP-h wrt. τ / W[1]-h wrt. k line graph NP-hard (Strict: FPT wrt. τ ) planar Take away message: Underlying graph restrictions help sometimes. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 10 / 23

  38. Complexity of Finding Temporal Separators First Summary We have seen so far: Hendrik Molter, TU Berlin On Separators in Temporal Graphs 11 / 23

  39. Complexity of Finding Temporal Separators First Summary We have seen so far: Layer restrictions: Hendrik Molter, TU Berlin On Separators in Temporal Graphs 11 / 23

  40. Complexity of Finding Temporal Separators First Summary We have seen so far: Layer restrictions: do not seem to help. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 11 / 23

  41. Complexity of Finding Temporal Separators First Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: Hendrik Molter, TU Berlin On Separators in Temporal Graphs 11 / 23

  42. Complexity of Finding Temporal Separators First Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 11 / 23

  43. Complexity of Finding Temporal Separators First Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Observation All these restrictions are invariant under reordering of layers! Hendrik Molter, TU Berlin On Separators in Temporal Graphs 11 / 23

  44. Complexity of Finding Temporal Separators First Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Observation All these restrictions are invariant under reordering of layers! Idea: Restrict “temporality” of the input graph. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 11 / 23

  45. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict p -monotone Definition ( cf. Khodaverdian et al. [2016]; Casteigts et al. [2012]) G = ( V , E 1 ,..., E τ ) is p -monotone if there are 1 = i 1 < ··· < i p + 1 = τ such that for all ℓ ∈ [ p ] it holds that E j ⊆ E j + 1 or E j ⊇ E j + 1 for all i ℓ ≤ j < i ℓ + 1 . Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  46. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 Definition ( cf. Khodaverdian et al. [2016]; Casteigts et al. [2012]) G = ( V , E 1 ,..., E τ ) is p -monotone if there are 1 = i 1 < ··· < i p + 1 = τ such that for all ℓ ∈ [ p ] it holds that E j ⊆ E j + 1 or E j ⊇ E j + 1 for all i ℓ ≤ j < i ℓ + 1 . Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  47. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 q -periodic Definition ( cf. Liu and Wu [2009]; Casteigts et al. [2012]; Flocchini et al. [2013]) G = ( V , E 1 ,..., E τ ) is q -periodic if E i = E i + q for all i ∈ [ τ − q ] . We call r := τ / q the number of periods. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  48. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 poly-time for q = 1, NP-h for q ≥ 1 NP-h for q ≥ 2 q -periodic Definition ( cf. Liu and Wu [2009]; Casteigts et al. [2012]; Flocchini et al. [2013]) G = ( V , E 1 ,..., E τ ) is q -periodic if E i = E i + q for all i ∈ [ τ − q ] . We call r := τ / q the number of periods. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  49. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 poly-time for q = 1, NP-h for q ≥ 1 NP-h for q ≥ 2 q -periodic poly-time if r ≥ n Definition ( cf. Liu and Wu [2009]; Casteigts et al. [2012]; Flocchini et al. [2013]) G = ( V , E 1 ,..., E τ ) is q -periodic if E i = E i + q for all i ∈ [ τ − q ] . We call r := τ / q the number of periods. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  50. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 Definition ( Kuhn et al. [2010]) poly-time for q = 1, NP-h for q ≥ 1 G = ( V , E 1 ,..., E τ ) is T -interval connected if for every NP-h for q ≥ 2 q -periodic t ∈ [ τ − T + 1 ] the graph G = ( V , ∩ t + T − 1 E i ) is connected. poly-time if r ≥ n i = t T -interval connected Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  51. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 Definition ( Kuhn et al. [2010]) poly-time for q = 1, NP-h for q ≥ 1 G = ( V , E 1 ,..., E τ ) is T -interval connected if for every NP-h for q ≥ 2 q -periodic t ∈ [ τ − T + 1 ] the graph G = ( V , ∩ t + T − 1 E i ) is connected. poly-time if r ≥ n i = t NP-h for T ≥ 1 NP-h for T ≥ 1 T -interval connected Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  52. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 Definition poly-time for q = 1, NP-h for q ≥ 1 G = ( V , E 1 ,..., E τ ) is λ -steady if for all t ∈ [ τ − 1 ] we have that NP-h for q ≥ 2 q -periodic | E t △ E t + 1 | ≤ λ . poly-time if r ≥ n NP-h for T ≥ 1 NP-h for T ≥ 1 T -interval connected λ -steady Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  53. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 Definition poly-time for q = 1, NP-h for q ≥ 1 G = ( V , E 1 ,..., E τ ) is λ -steady if for all t ∈ [ τ − 1 ] we have that NP-h for q ≥ 2 q -periodic | E t △ E t + 1 | ≤ λ . poly-time if r ≥ n NP-h for T ≥ 1 NP-h for T ≥ 1 T -interval connected poly-time for λ = 0, λ -steady NP-h for λ ≥ 0 NP-h for λ ≥ 1 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  54. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 poly-time for q = 1, NP-h for q ≥ 1 NP-h for q ≥ 2 q -periodic poly-time if r ≥ n NP-h for T ≥ 1 NP-h for T ≥ 1 T -interval connected poly-time for λ = 0, λ -steady NP-h for λ ≥ 0 NP-h for λ ≥ 1 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  55. Complexity of Finding Temporal Separators Temporal Restrictions Temporal graph classes with temporal aspects: ( s , z ) -Separation Restriction Strict Non-Strict poly-time for p = 1, p -monotone NP-h for p ≥ 1 NP-h for p ≥ 2 poly-time for q = 1, NP-h for q ≥ 1 NP-h for q ≥ 2 q -periodic poly-time if r ≥ n NP-h for T ≥ 1 NP-h for T ≥ 1 T -interval connected poly-time for λ = 0, λ -steady NP-h for λ ≥ 0 NP-h for λ ≥ 1 Hendrik Molter, TU Berlin On Separators in Temporal Graphs 12 / 23

  56. Complexity of Finding Temporal Separators Second Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 13 / 23

  57. Complexity of Finding Temporal Separators Second Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Temporal restrictions: Hendrik Molter, TU Berlin On Separators in Temporal Graphs 13 / 23

  58. Complexity of Finding Temporal Separators Second Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Temporal restrictions: do not seem to help. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 13 / 23

  59. Complexity of Finding Temporal Separators Second Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Temporal restrictions: do not seem to help. Idea: Tailored restrictions that do not fit into the above categories. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 13 / 23

  60. Complexity of Finding Temporal Separators Second Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Temporal restrictions: do not seem to help. Idea: Tailored restrictions that do not fit into the above categories. Order-Preserving Temporal Unit Interval Graphs. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 13 / 23

  61. Complexity of Finding Temporal Separators Second Summary We have seen so far: Layer restrictions: do not seem to help. Underlying graph restrictions: help only in few cases. Temporal restrictions: do not seem to help. Idea: Tailored restrictions that do not fit into the above categories. Order-Preserving Temporal Temporal Graph with Unit Interval Graphs. bounded-sized Temporal Core. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 13 / 23

  62. ( s , z ) -Separation on Temporal Unit Interval Graphs Order-Preserving Temporal Unit Interval Graph Order-Preserving Temporal Unit Interval Graph A temporal graph G = ( V , E 1 ,..., E τ ) is an order-preserving temporal unit interval graph if Hendrik Molter, TU Berlin On Separators in Temporal Graphs 14 / 23

  63. ( s , z ) -Separation on Temporal Unit Interval Graphs Order-Preserving Temporal Unit Interval Graph Order-Preserving Temporal Unit Interval Graph A temporal graph G = ( V , E 1 ,..., E τ ) is an order-preserving temporal unit interval graph if each layer is a unit interval graph, and Hendrik Molter, TU Berlin On Separators in Temporal Graphs 14 / 23

  64. ( s , z ) -Separation on Temporal Unit Interval Graphs Order-Preserving Temporal Unit Interval Graph Order-Preserving Temporal Unit Interval Graph A temporal graph G = ( V , E 1 ,..., E τ ) is an order-preserving temporal unit interval graph if each layer is a unit interval graph, and there is a total ordering < V which is compatible with each layer. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 14 / 23

  65. ( s , z ) -Separation on Temporal Unit Interval Graphs Order-Preserving Temporal Unit Interval Graph Order-Preserving Temporal Unit Interval Graph A temporal graph G = ( V , E 1 ,..., E τ ) is an order-preserving temporal unit interval graph if each layer is a unit interval graph, and there is a total ordering < V which is compatible with each layer. Recall: < V is compatible with a unit interval graph G = ( V , E ) if { x , y } ∈ E with x < V y implies { v ∈ V | x ≤ V v ≤ V y } is a clique. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 14 / 23

  66. ( s , z ) -Separation on Temporal Unit Interval Graphs Order-Preserving Temporal Unit Interval Graph Order-Preserving Temporal Unit Interval Graph A temporal graph G = ( V , E 1 ,..., E τ ) is an order-preserving temporal unit interval graph if each layer is a unit interval graph, and there is a total ordering < V which is compatible with each layer. Recall: < V is compatible with a unit interval graph G = ( V , E ) if { x , y } ∈ E with x < V y implies { v ∈ V | x ≤ V v ≤ V y } is a clique. Motivation: Physical proximity networks in one-dimensional spaces. Hendrik Molter, TU Berlin On Separators in Temporal Graphs 14 / 23

  67. ( s , z ) -Separation on Temporal Unit Interval Graphs Poly-time Algo for Non-Strict ( s , z ) -Separation Order-Preserving Temporal Unit Interval Graphs v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 s z Vertex Ordering < V Hendrik Molter, TU Berlin On Separators in Temporal Graphs 15 / 23

  68. ( s , z ) -Separation on Temporal Unit Interval Graphs Poly-time Algo for Non-Strict ( s , z ) -Separation Order-Preserving Temporal Unit Interval Graphs v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 s z Vertex Ordering < V Hendrik Molter, TU Berlin On Separators in Temporal Graphs 15 / 23

  69. ( s , z ) -Separation on Temporal Unit Interval Graphs Poly-time Algo for Non-Strict ( s , z ) -Separation Order-Preserving Temporal Unit Interval Graphs v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 s z Vertex Ordering < V Hendrik Molter, TU Berlin On Separators in Temporal Graphs 15 / 23

  70. ( s , z ) -Separation on Temporal Unit Interval Graphs Poly-time Algo for Non-Strict ( s , z ) -Separation Order-Preserving Temporal Unit Interval Graphs v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 s z Vertex Ordering < V Hendrik Molter, TU Berlin On Separators in Temporal Graphs 15 / 23

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