DMGT

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Discussiones Mathematicae Graph Theory

ON NON-z(mod k) DOMINATING SETS

 Yair Caro Department of Mathematics University of Haifa - Oranim Tivon - 36006, ISRAEL e-mail: ya_caro@kvgeva.org.il Michael S. Jacobson Department of Mathematics University of Louisville Louisville, KY 40292, USA e-mail: mikej@louisville.edu

Abstract

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.

2000 Mathematics Subject Classification: 05C69, 05C85.

References

 [1] A. Amin and P. Slater, Neighborhood Domination with Parity Restriction in Graphs, Congr. Numer. 91 (1992) 19-30. [2] A. Amin and P. Slater, All Parity Realizable Trees, J. Combin. Math. Combin. Comput. 20 (1996) 53-63. [3] Y. Caro, Simple Proofs to Three Parity Theorems, Ars Combin. 42 (1996) 175-180. [4] Y. Caro and W.F. Klostermeyer, The odd domination number of a graph, to appear in J. Combin. Math. Combin. Comput. [5] Y. Caro, J. Goldwasser and W. Klostermeyer, Odd and Residue Domination Numbers of a Graph, Discuss. Math. Graph Theory 21 (2001) 119-136, doi: 10.7151/dmgt.1137. [6] J. Goldwasser and W. Klostermeyer, Maximization Versions of Lights Out Games in Grids and Graphs, Congr. Numer. 126 (1997) 99-111. [7] K. Sutner, Linear Cellular Automata and the Garden-of-Eden, Mathematical Intelligencer 11 (2) (1989) 49-53.

Received 26 November 2001
Revised 26 February 2002