Minimum k-Hop Dominating Sets in Grid Graphs
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Given a graph G, the k-hop dominating set problem asks for a vertex subset D_k such that every vertex of G is in distance at most k to some vertex in D_k (k∈ℕ). For k=1, this corresponds to the classical dominating set problem in graphs. We study the k-hop dominating set problem in grid graphs (motivated by generalized guard sets in polyominoes). We show that the VC dimension of this problem is 3 in grid graphs without holes, and 4 in general grid graphs. Furthermore, we provide a reduction from planar monotone 3SAT, thereby showing that the problem is NP-complete even in thin grid graphs (i.e., grid graphs that do not a contain an induced C_4). Complementary, we present a linear-time 4-approximation algorithm for 2-thin grid graphs (which do not contain a 3× 3-grid subgraph) for all k∈ℕ.
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