### Abstract

In 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.

Original language | English |
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Pages (from-to) | 822-848 |

Journal | Theory of Computing Systems |

Volume | 45 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

## Cite this

Fellows, M. R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F., & Saurabh, S. (2009). The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number.

*Theory of Computing Systems*,*45*(4), 822-848. https://doi.org/10.1007/s00224-009-9167-9