Fellows, MichaelMichaelFellowsLokshtanov, DanielDanielLokshtanovMisra, NeeldharaNeeldharaMisraMnich, MatthiasMatthiasMnichRosamond, FrancesFrancesRosamondSaurabh, SaketSaketSaurabh2020-01-242020-01-242009-01-09Theory of Computing Systems (2009).http://hdl.handle.net/11420/4557In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W[1]-hard under this parameterization? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G)≤k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O *(f(k)). (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.en1433-04902009822848Springer NatureInformatikJournal Article10.1007/s00224-009-9167-9Other