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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1195,
author = {Yair Caro and Michael S. Jacobson},
title = {On non-z(mod k) dominating sets},
journal = {Discussiones Mathematicae Graph Theory},
volume = {23},
year = {2003},
pages = {189-199},
zbl = {1050.05092},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1195}
}
Yair Caro; Michael S. Jacobson. On non-z(mod k) dominating sets. Discussiones Mathematicae Graph Theory, Tome 23 (2003) pp. 189-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1195/
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