Geodesic
On non-z(mod k) dominating sets
Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 189-199

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For a graph G, a positive integer k, k ≥ 2, and a non-negative integer with z k and z ≠ 1, a subset D of the vertex set V(G) is said to be a non-z (mod k) dominating set if D is a dominating set and for all x ∈ V(G), |N[x]∩D| ≢ z (mod k).For the case k = 2 and z = 0, it has been shown that these sets exist for all graphs. The problem for k ≥ 3 is unknown (the existence for even values of k and z = 0 follows from the k = 2 case.) It is the purpose of this paper to show that for k ≥ 3 and with z k and z ≠ 1, that a non-z(mod k) dominating set exist for all trees. Also, it will be shown that for k ≥ 4, z ≥ 1, 2 or 3 that any unicyclic graph contains a non-z(mod k) dominating set. We also give a few special cases of other families of graphs for which these dominating sets must exist.
Keywords: dominating set, tree, unicyclic graph 
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Caro, Yair; Jacobson, Michael. On non-z(mod k) dominating sets. Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 189-199. http://geodesic.mathdoc.fr/item/DMGT_2003_23_1_a13/