Abstract
We consider the problem \(\mathrm {LIS}\) of deciding whether there exists an induced subtree with exactly \(i \le n\) vertices and \(\ell \) leaves in a given graph G with n vertices. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by \(L_G(i)\), realized by an induced subtree with i vertices, for \(0 \le i \le n\). We begin by proving that the \(\mathrm {LIS}\) problem is NP-complete in general. Then, we describe a nontrivial branch and bound algorithm that computes the function \(L_G\) for any simple graph G. In the special case where G is a tree of maximum degree \(\varDelta \), we provide a \(\mathcal {O}(n^3\varDelta )\) time and \(\mathcal {O}(n^2)\) space algorithm to compute the function \(L_G\).
A. Blondin Massé is supported by a grant from the National Sciences and Engineering Research Council of Canada (NSERC) through Individual Discovery Grant RGPIN-417269-2013. M. Lapointe and É. Nadeau are both supported by a scholarship from the NSERC.
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Blondin Massé, A., de Carufel, J., Goupil, A., Lapointe, M., Nadeau, É., Vandomme, É. (2018). Fully Leafed Induced Subtrees. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_8
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