. The Parameterized Complexity of Graph Cyclability .... .... .... - PowerPoint PPT Presentation

. .... .... .... .... .... .... . The Parameterized Complexity of Graph Cyclability .... .... .... .... . . Joint work with Petr A. Golovach, Marcin Kamiński, and Dimitrios M. Thilikos AGTAC, Koper, June 2015 Spyridon Maniatis .... .. .. ... . ... . ... . ... . .. .. .. . . .. .. .. .

. .. .. . . .. . . . . . .. . . .. . . .. .. . UoA Spyridon Maniatis which G is k -cyclable. . . Cyclability (or a way to unify connectivity and Hamiltonicity): . Can be thought as of a quantitive measure of Hamiltonicity We study (from the algorithmic point of view) a connectivity Introduction . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability related parameter, namely cyclability [V. Chvátal, 1973]. A graph G is k -cyclable if every k vertices of V ( G ) lie in a common cycle. The cyclability of G is the maximum integer k for

. .. .. . . .. . . . . . .. . . .. . . .. .. . UoA Spyridon Maniatis which G is k -cyclable. . . Cyclability (or a way to unify connectivity and Hamiltonicity): . Can be thought as of a quantitive measure of Hamiltonicity We study (from the algorithmic point of view) a connectivity Introduction . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability related parameter, namely cyclability [V. Chvátal, 1973]. A graph G is k -cyclable if every k vertices of V ( G ) lie in a common cycle. The cyclability of G is the maximum integer k for

. .. .. . . .. . . . . . .. . . .. . . .. .. . UoA Spyridon Maniatis which G is k -cyclable. . . Cyclability (or a way to unify connectivity and Hamiltonicity): . Can be thought as of a quantitive measure of Hamiltonicity We study (from the algorithmic point of view) a connectivity Introduction . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability related parameter, namely cyclability [V. Chvátal, 1973]. A graph G is k -cyclable if every k vertices of V ( G ) lie in a common cycle. The cyclability of G is the maximum integer k for

. . .. . . .. . . .. . . .. . . .. . .. . Natural question : Is there an efficient (polynomial?) UoA Spyridon Maniatis From the parameterized complexity point of view? cubic planar graphs). NO, because hamiltonian cycle is NP -hard (even for algorithm computing the cyclability of a graph? Introduction . . .. . . .. . . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . .. . .. . . . .. . . .. The Parameterized Complexity of Graph Cyclability

. . .. . . .. . . .. . . .. . . .. . .. . Natural question : Is there an efficient (polynomial?) UoA Spyridon Maniatis From the parameterized complexity point of view? cubic planar graphs). NO, because hamiltonian cycle is NP -hard (even for algorithm computing the cyclability of a graph? Introduction . . .. . . .. . . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . .. . .. . . . .. . . .. The Parameterized Complexity of Graph Cyclability

. . .. . . .. . . .. . . .. . . .. . .. . Natural question : Is there an efficient (polynomial?) UoA Spyridon Maniatis From the parameterized complexity point of view? cubic planar graphs). NO, because hamiltonian cycle is NP -hard (even for algorithm computing the cyclability of a graph? Introduction . . .. . . .. . . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . .. . .. . . . .. . . .. The Parameterized Complexity of Graph Cyclability

. .. .. . . .. . . . . . .. . . .. . . .. .. . UoA Spyridon Maniatis annotated version of the problem: We actually consider, for technical reasons, the more general, Input: A graph G and a positive integer k . . . . p - Cyclability . . The Parameterized Problem . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability Parameter: k . Question: Is G k -cyclable?

. .. . . .. . . .. . . .. . . .. . . . . p - Annotated Cyclability . UoA Spyridon Maniatis problem. . . . . . The Annotated Version . .. . . .. .. .. . . .. . .. . . .. . . .. . . .. . . .. . . . . . .. . .. .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability Input: A graph G , a set R ⊆ V ( G ) and a positive integer k . Parameter: k . Question: Is it true that for every S ⊆ R with | S | ≤ k , there exists a cycle of G that meets all the vertices of S ? Of course, when R = V ( G ) we have an instance of the initial

. .. . . .. . . .. . . .. . . .. . . . . p - Annotated Cyclability . UoA Spyridon Maniatis problem. . . . . . The Annotated Version . .. . . .. .. .. . . .. . .. . . .. . . .. . . .. . . .. . . . . . .. . .. .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability Input: A graph G , a set R ⊆ V ( G ) and a positive integer k . Parameter: k . Question: Is it true that for every S ⊆ R with | S | ≤ k , there exists a cycle of G that meets all the vertices of S ? Of course, when R = V ( G ) we have an instance of the initial

. .. . . .. . . .. . . .. . . .. . . . .. results are: UoA Spyridon Maniatis restricted to cubic planar graphs. graphs. 2 The problem is in FPT when restricted to the class of planar 1 Cyclability is co-W[1] -hard (even for split-graphs), when We study the parameterized complexity of Cyclability . Our . Our results . .. . . .. .. . . . . . .. . . .. . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability parameterized by k . 3 No polynomial kernel unless NP ⊆ co-NP/poly , when

. .. . . .. . . .. . . .. . . .. . . . .. results are: UoA Spyridon Maniatis restricted to cubic planar graphs. graphs. 2 The problem is in FPT when restricted to the class of planar 1 Cyclability is co-W[1] -hard (even for split-graphs), when We study the parameterized complexity of Cyclability . Our . Our results . .. . . .. .. . . . . . .. . . .. . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability parameterized by k . 3 No polynomial kernel unless NP ⊆ co-NP/poly , when

. .. . . .. . . .. . . .. . . .. . . . .. results are: UoA Spyridon Maniatis restricted to cubic planar graphs. graphs. 2 The problem is in FPT when restricted to the class of planar 1 Cyclability is co-W[1] -hard (even for split-graphs), when We study the parameterized complexity of Cyclability . Our . Our results . .. . . .. .. . . . . . .. . . .. . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability parameterized by k . 3 No polynomial kernel unless NP ⊆ co-NP/poly , when

. .. . . .. . . .. . . .. . . .. . . . .. results are: UoA Spyridon Maniatis restricted to cubic planar graphs. graphs. 2 The problem is in FPT when restricted to the class of planar 1 Cyclability is co-W[1] -hard (even for split-graphs), when We study the parameterized complexity of Cyclability . Our . Our results . .. . . .. .. . . . . . .. . . .. . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . The Parameterized Complexity of Graph Cyclability parameterized by k . 3 No polynomial kernel unless NP ⊆ co-NP/poly , when

. . . .. . . .. . .. .. . . .. . . .. . . . .. . UoA Spyridon Maniatis G that contains all vertices of S ? Parameter: k . Input: A split graph G and a positive integer k . . . p - Cyclability complement . Theorem 1 . the inputs are restricted to be split graphs. The p - Cyclability problem is co-W[1] -hard. This also holds if . . Theorem 1 . . .. . .. . .. . . .. . . . . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. The Parameterized Complexity of Graph Cyclability Reduction of the k - Clique problem to: Question: Is there an S ⊆ V ( G ) , | S | ≤ k s.t. there is no cycle of

. . . . .. . . .. . .. . . . .. . . .. . .. Theorem 1 .. . UoA Spyridon Maniatis G that contains all vertices of S ? Parameter: k . Input: A split graph G and a positive integer k . . . p - Cyclability complement . . . Reduction of the k - Clique problem to: the inputs are restricted to be split graphs. The p - Cyclability problem is co-W[1] -hard. This also holds if . . Theorem 1 . .. . .. . .. . . .. . . . . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. The Parameterized Complexity of Graph Cyclability Question: Is there an S ⊆ V ( G ) , | S | ≤ k s.t. there is no cycle of