Planar Capacitated Dominating Set Is W[1]-Hard

Authors:
Hans L. Bodlaender;Daniel Lokshtanov;Eelko Penninkx
Affiliations:
Department of Information and Computing Sciences, Universiteit Utrecht, Utrecht, The Netherlands 3508TB;Department of Informatics, University of Bergen, Bergen, Norway N-5020;Department of Information and Computing Sciences, Universiteit Utrecht, Utrecht, The Netherlands 3508TB
Venue:
Parameterized and Exact Computation
Year:
2009

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Abstract

Given a graph G together with a capacity function c : V(G) 驴驴, we call S 驴 V(G) a capacitated dominating set if there exists a mapping f: (V(G) 驴 S) 驴S which maps every vertex in (V(G) 驴 S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v 驴 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.