PERFECT DOMINATING SETS

A dominating set of a graph is perfect if each vertex of is dominated by exactly one vertex in . We study the existence and construction of PDSs in families of graphs arising from the interconnection networks of parallel computers. These inc lude trees, dags, series-parallel graphs, meshes, tori, hypercubes, cube-connected cycles, cube-connected paths, and de Bruijn graphs. For trees, dags, and series-parallel graphs we give linear t ime algorithms that determine if a PDS exists, and generate a PDS when one does. For 2- and 3-dimensional meshes, 2-dimensional tori, hypercubes, and cube-connected paths we completely characterize which graphs have a PDS, and the structure of all PDSs. For higher dimensional meshes and tori, cube-connected cycles, and de Bruijn graphs, we show the existence of a PDS in infinitely man y cases, but our characterization is not complete. Our results include distance -domination for arbitrary .

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