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• # Artykuł - szczegóły

## Open Mathematics

2014 | 12 | 3 | 517-528

## Signed k-independence in graphs

EN

### Abstrakty

EN
Let k ≥ 2 be an integer. A function f: V(G) → {−1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v] f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α sk(G) of G. In this work, we mainly present upper bounds on α sk (G), as for example α sk(G) ≤ n − 2⌈(Δ(G) + 2 − k)/2⌉, and we prove the Nordhaus-Gaddum type inequality $$\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$$, where n is the order, Δ(G) the maximum degree and $$\bar G$$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.

EN

517-528

wydano
2014-03-01
online
2013-12-21

### Twórcy

autor
• RWTH-Aachen University

### Bibliografia

• [1] Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals of Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 208, Marcel Dekker, New York, 1998
• [2] Haynes T.W., Hedetniemi S.T., Slater P.J. (Eds.), Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 209, Marcel Dekker, New York, 1998
• [3] Henning M.A., Signed 2-independence in graphs, Discrete Math., 2002, 250(1–3), 93–107 http://dx.doi.org/10.1016/S0012-365X(01)00275-8
• [4] Shan E., Sohn M.Y., Kang L., Upper bounds on signed 2-independence numbers of graphs, Ars Combin., 2003, 69, 229–239
• [5] Turán P., On an extremal problem in graph theory, Mat. Fiz. Lapok, 1941, 48, 436–452 (in Hungarian)
• [6] Volkmann L., Bounds on the signed 2-independence number in graphs, Discuss. Math. Graph Theory, 2013, 33(4), 709–715 http://dx.doi.org/10.7151/dmgt.1686
• [7] Zelinka B., On signed 2-independence numbers of graphs (manuscript)