# Signed k-independence in graphs

Open Mathematics (2014)

• Volume: 12, Issue: 3, page 517-528
• ISSN: 2391-5455

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## Abstract

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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 ${\alpha }_{S}^{k}\left(G\right)+{\alpha }_{S}^{k}\left(\overline{G}\right)⩽n+2k-3$ , where n is the order, Δ(G) the maximum degree and $\overline{G}$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.

## How to cite

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Lutz Volkmann. "Signed k-independence in graphs." Open Mathematics 12.3 (2014): 517-528. <http://eudml.org/doc/269006>.

@article{LutzVolkmann2014,
abstract = {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.},
author = {Lutz Volkmann},
journal = {Open Mathematics},
keywords = {Bounds; Signed k-independence function; Signed k-independence number; Nordhaus-Gaddum type result; bounds; signed -independence function; signed -independence number},
language = {eng},
number = {3},
pages = {517-528},
title = {Signed k-independence in graphs},
url = {http://eudml.org/doc/269006},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Lutz Volkmann
TI - Signed k-independence in graphs
JO - Open Mathematics
PY - 2014
VL - 12
IS - 3
SP - 517
EP - 528
AB - 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.
LA - eng
KW - Bounds; Signed k-independence function; Signed k-independence number; Nordhaus-Gaddum type result; bounds; signed -independence function; signed -independence number
UR - http://eudml.org/doc/269006
ER -

## References

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1. [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 Zbl0890.05002
2. [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 Zbl0890.05002
3. [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 Zbl1003.05076
4. [4] Shan E., Sohn M.Y., Kang L., Upper bounds on signed 2-independence numbers of graphs, Ars Combin., 2003, 69, 229–239 Zbl1073.05559
5. [5] Turán P., On an extremal problem in graph theory, Mat. Fiz. Lapok, 1941, 48, 436–452 (in Hungarian)
6. [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 Zbl1295.05182
7. [7] Zelinka B., On signed 2-independence numbers of graphs (manuscript)

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