Planar Capacitated Dominating Set Is W[1]-Hard

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.