TY  - JOUR
A1  - Casel, Katrin
A1  - Dreier, Jan
A1  - Fernau, Henning
A1  - Gobbert, Moritz
A1  - Kuinke, Philipp
A1  - Villaamil, Fernando Sánchez
A1  - Schmid, Markus L.
A1  - van Leeuwen, Erik Jan
T1  - Complexity of independency and cliquy trees
JF  - Discrete applied mathematics
N2  - An independency (cliquy) tree of an n-vertex graph G is a spanning tree of G in which the set of leaves induces an independent set (clique). We study the problems of minimizing or maximizing the number of leaves of such trees, and fully characterize their parameterized complexity. We show that all four variants of deciding if an independency/cliquy tree with at least/most l leaves exists parameterized by l are either Para-NP- or W[1]-hard. We prove that minimizing the number of leaves of a cliquy tree parameterized by the number of internal vertices is Para-NP-hard too. However, we show that minimizing the number of leaves of an independency tree parameterized by the number k of internal vertices has an O*(4(k))-time algorithm and a 2k vertex kernel. Moreover, we prove that maximizing the number of leaves of an independency/cliquy tree parameterized by the number k of internal vertices both have an O*(18(k))-time algorithm and an O(k 2(k)) vertex kernel, but no polynomial kernel unless the polynomial hierarchy collapses to the third level. Finally, we present an O(3(n) . f(n))-time algorithm to find a spanning tree where the leaf set has a property that can be decided in f (n) time and has minimum or maximum size.
KW  - independency tree
KW  - cliquy tree
KW  - parameterized complexity
KW  - Kernelization
KW  - algorithms
KW  - exact algorithms
Y1  - 2018
U6  - https://doi.org/10.1016/j.dam.2018.08.011
SN  - 0166-218X
SN  - 1872-6771
VL  - 272
SP  - 2
EP  - 15
PB  - Elsevier
CY  - Amsterdam [u.a.]
ER  -