Exact Elimination of Cycles in Graphs

One of the standard basic steps in drawing hierarchical graphs is to invert some arcs of the given graph to make the graph acyclic. We discuss exact and parameterized algorithms for this problem. In particular we examine a graph class called $(1,n)$-graphs, which contains cubic graphs. For both exact and parameterized algorithms we use a non-standard measure approach for the analysis. The analysis of the parameterized algorithm is of special interest, as it is not an amortized analysis modelled by 'finite states' but rather a 'top-down' amortized analysis. For $(1,n)$-graphs we achieve a running time of $Oh^*(1.1871^m)$ and $Oh^*(1.212^k)$, for cubic graphs $Oh^*(1.1798^m)$ and $Oh^*(1.201^k)$, respectively. As a by-product the trivial bound of $2^n$ for {sc Feedback Vertex Set} on planar directed graphs is broken.