Abstract
Given a graph G together with a capacity function c : V(G) →ℕ, we call S ⊆ V(G) a capacitated dominating set if there exists a mapping f: (V(G) ∖ S) →S which maps every vertex in (V(G) ∖ S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a capacity function c and a positive integer k and asked whether G has a capacitated dominating set of size at most k. In this paper we show that Planar Capacitated Dominating Set is W[1]-hard, resolving an open problem of Dom et al. [IWPEC, 2008 ]. This is the first bidimensional problem to be shown W[1]-hard. Thus Planar Capacitated Dominating Set can become a useful starting point for reductions showing parameterized intractablility of planar graph problems.
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Bodlaender, H.L., Lokshtanov, D., Penninkx, E. (2009). Planar Capacitated Dominating Set Is W[1]-Hard. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_4
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DOI: https://doi.org/10.1007/978-3-642-11269-0_4
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