Complexity of independency and cliquy trees

Abstract

An independency (cliquy) tree of an -vertex graph is a spanning tree of 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 leaves exists parameterized by are either - or -hard. We prove that minimizing the number of leaves of a cliquy tree parameterized by the number of internal vertices is -hard too. However, we show that minimizing the number of leaves of an independency tree parameterized by the number of internal vertices has an -time algorithm and a vertex kernel. Moreover, we prove that maximizing the number of leaves of an independency/cliquy tree parameterized by the number of internal vertices both have an -time algorithm and an vertex kernel, but no polynomial kernel unless the polynomial hierarchy collapses to the third level. Finally, we present an -time algorithm to find a spanning tree where the leaf set has a property that can be decided in time and has minimum or maximum size.