Abstract
We investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems, namely k-Vertex Cover and k-Edge Cover. Specifically, we impose the additional requirement that each connected component of a solution have at least t vertices (resp. edges from the solution), and call the problem t-total vertex cover (resp. t-total edge cover). We show that
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both problems remain fixed-parameter tractable with these restrictions, with running times of the form \({\mathcal O}^{*}\left(c^{k}\right)\) for some constant c > 0 in each case;
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for every t ≥ 2, t-total vertex cover has no polynomial kernel unless the Polynomial Hierarchy collapses to the third level;
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for every t ≥ 2, t-total edge cover has a linear vertex kernel of size \(\frac{t+1}{t}k\).
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Fernau, H., Fomin, F.V., Philip, G., Saurabh, S. (2010). The Curse of Connectivity: t-Total Vertex (Edge) Cover. In: Thai, M.T., Sahni, S. (eds) Computing and Combinatorics. COCOON 2010. Lecture Notes in Computer Science, vol 6196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14031-0_6
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DOI: https://doi.org/10.1007/978-3-642-14031-0_6
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