Abstract
Given a positive integer \(k\) and a graph \(G\), a \(k\)-limited packing in \(G\) (2010) is a subset \(B\) of its vertex set such that each closed neighborhood has at most \(k\) vertices of \(B\). As a variation, we introduce the notion of a \(\{k\}\)-packing function \(f\) of \(G\) which assigns a non-negative integer to the vertices of \(G\) in such a way that the sum of \(f(v)\) over each closed neighborhood is at most \(k\). For fixed \(k\), we prove that the problem of finding a \(\{k\}\)-packing function of maximum weight (\(\{k\)}PF) can be reduced linearly to the problem of finding a \(k\)-limited packing of maximum cardinality (\(k\)LP). We present an \(O(|V(G)|+|E(G)|)\) time algorithm to solve \(\{k\)}PF on strongly chordal graphs. We also use monadic second-order logic to prove that both problems are linear time solvable for graphs with clique-width bounded by a constant.
Partially supported by grantsPICT ANPCyT 0482 (2011–2013) and 1ING 391 (2012–2014).
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Acknowledgments
We are grateful to H. Freytes for the discussions held with us around first- and second-order logics.
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Leoni, V.A., Hinrichsen, E.G. (2014). \(\{k\}\)-Packing Functions of Graphs. In: Fouilhoux, P., Gouveia, L., Mahjoub, A., Paschos, V. (eds) Combinatorial Optimization. ISCO 2014. Lecture Notes in Computer Science(), vol 8596. Springer, Cham. https://doi.org/10.1007/978-3-319-09174-7_28
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DOI: https://doi.org/10.1007/978-3-319-09174-7_28
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