(i) a kernel with at most αk vertices and βk edges for MAXCUT. Here 0<α1 and 1<β2. Values of α and β depends on the number of vertices and the edges in the graph;

(ii) a kernel with at most 4k/3 vertices and 2k edges for MAXDAG;

(iii) an O^*(1.2418^k) parameterized algorithm for MAXCUT in undirected graphs. This improves the O^*(1.4143^k)¹ algorithm presented in [E. Prieto, The method of extremal structure on the k-maximum cut problem, in: The Proceedings of Computing: The Australasian Theory Symposium (CATS), 2005, pp. 119–126];

(iv) an O^*(2ⁿ) algorithm for optimization version of MAXDAG in directed graphs. This is the first such algorithm to the best of our knowledge;

(v) an O^*(2^k) parameterized algorithm for MAXDAG in directed graphs. This improves the previous best of O^*(4^k) presented in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446–458];

(vi) an O^*(16^k) parameterized algorithm to determine whether an oriented graph having m arcs has an acyclic subgraph with at least m/2+k arcs. This improves the O^*(2^k) algorithm given in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446–458].

In addition, we show that if a directed graph has minimum out degree at least f(n) (some function of n) then Directed Feedback Arc Set problem is fixed parameter tractable. The parameterized complexity of Directed Feedback Arc Set is a well-known open problem.