Abstract
We consider the \(\mathcal{NP}\)-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O *(c n) (c ≤ 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8669n) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364k n O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.
This work was partially Supported by a PPP grant between DAAD (Germany) and NFR (Norway).
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Fernau, H., Gaspers, S., Raible, D. (2010). Exact and Parameterized Algorithms for Max Internal Spanning Tree . In: Paul, C., Habib, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2009. Lecture Notes in Computer Science, vol 5911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11409-0_9
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DOI: https://doi.org/10.1007/978-3-642-11409-0_9
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