Geodesic
Efficient (j,k)-domination
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 409-423.

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A dominating set S of a graph G is called efficient if |N[v]∩ S| = 1 for every vertex v ∈ V(G). That is, a dominating set S is efficient if and only if every vertex is dominated exactly once. In this paper, we investigate efficient multiple domination. There are several types of multiple domination defined in the literature: k-tuple domination, k-domination, and k-domination. We investigate efficient versions of the first two as well as a new type of multiple domination.
Keywords: efficient domination, multiple domination 
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Rubalcaba, Robert; Slater, Peter. Efficient (j,k)-domination. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 409-423. http://geodesic.mathdoc.fr/item/DMGT_2007_27_3_a2/

[1] D.W. Bange, A.E. Barkauskas and P.J. Slater, Disjoint dominating sets in trees, Sandia Laboratories Report, SAND 78-1087J (1978).

[2] D.W. Bange, A.E. Barkauskas and P.J. Slater, Efficient dominating sets in graphs, in: R.D. Ringeisen and F.S. Roberts (eds.) Applications of Discrete Mathematics, pages 189-199, (SIAM, Philadelphia, PA, 1988).

[3] D.W. Bange, A.E. Barkauskas and P.J. Slater, Efficient near-domination of grid graphs, Congr. Numer. 58 (1986) 83-92.

[4] D.W. Bange, A.E. Barkauskas, L.H. Host and P.J. Slater, Generalized domination and efficient domination in graphs, Discrete Math. 159 (1996) 1-11, doi: 10.1016/0012-365X(95)00094-D.

[5] N. Biggs, Perfect codes in graphs, J. Combin. Theory (B) 15 (1973) 288-296, doi: 10.1016/0095-8956(73)90042-7.

[6] M. Chellali, A. Khelladi and F. Maffray, Exact double domination in graphs, Discuss. Math. Graph Theory 25 (2005) 291-302, doi: 10.7151/dmgt.1282.

[7] G.S. Domke, S.T. Hedetniemi, R.C. Laskar and G. Fricke, Relationships between integer and fractional parameters of graphs, in: Y. Alavi, G. Chartrand, O.R. Oellermann and A.J. Schwenk (eds.), Graph Theory, Combinatorics, and Applications, Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, vol. 1, pages 371-387, (Kalamazoo, MI 1988), Wiley Publications, 1991.

[8] M. Farber, Domination, independent domination, and duality in strongly chordal graphs, Discrete Appl. Math. 7 (1984) 115-130, doi: 10.1016/0166-218X(84)90061-1.

[9] J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science (Wiley, New York, 1984) 283-300.

[10] W. Goddard and M.A. Henning, Real and integer domination in graphs, Discrete Math. 199 (1999) 61-75, doi: 10.1016/S0012-365X(98)00286-6.

[11] D.L. Grinstead and P.J. Slater, On the minimum intersection of minimum dominating sets in series-parallel graphs, in: Y. Alavi, G. Chartrand, O.R. Oellermann and A.J. Schwenk (eds.), Graph Theory, Combinatorics, and Applications, Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, vol. 1, pages 563-584, (Kalamazoo, MI 1988), Wiley Publications, 1991.

[12] D.L. Grinstead and P.J. Slater, Fractional domination and fractional packing in graphs, Congr. Numer. 71 (1990) 153-172.

[13] F. Harary and T.W. Haynes, Nordhaus-Gaddum inequalities for domination in graphs, Discrete Mathematics 155 (1996) 99-105, doi: 10.1016/0012-365X(94)00373-Q.

[14] R.R. Rubalcaba and M. Walsh, Minimum fractional dominating and maximum fractional packing functions, submitted.

[15] R.R. Rubalcaba and P.J. Slater, A note on obtaining k dominating sets from a k-dominating function on a tree, submitted.

[16] P.J. Slater, Generalized graph parameters: Gallai theorems I, Bull. Inst. of Comb. Appl. 17 (1996) 27-37.