Discussiones Mathematicae Graph Theory 33(4) (2013)
709-715

doi: 10.7151/dmgt.1686

Lutz Volkmann
Lehrstuhl II für Mathematik |

In this work, we mainly present upper bounds on α_{s}^{2}(G), as for example
α_{s}^{2}(G) ≤ n −2 ⌈ Δ(G)/2 ⌉, and we prove the Nordhaus-Gaddum type inequality
α_{s}^{2}(G)+ α_{s}^{2}(G) ≤ n+1, where n is the order and Δ(G) is the maximum
degree of the graph G. Some of our theorems improve well-known results on the signed 2-independence
number.

**Keywords:** bounds, signed 2-independence function, signed 2-independence number, Nordhaus-Gaddum type result

**2010 Mathematics Subject Classification:** 05C69.

[1] | J.E. Dunbar, S.T. Hedetniemi, M.A. Henning and P.J. Slater, Signed domination in graphs, in: Graph Theory, Combinatorics, and Applications (John Wiley and Sons, Inc. 1, 1995) 311--322. |

[2] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs ( Marcel Dekker, Inc., New York, 1998). |

[3] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics ( Marcel Dekker, Inc., New York, 1998). |

[4] | M.A. Henning, Signed , Discrete Math. 2-independence in graphs 250 (2002) 93--107, doi: 10.1016/S0012-365X(01)00275-8. |

[5] | E.F. Shan, M.Y. Sohn and L.Y. Kang, Upper bounds on signed , Ars Combin. 2-independence numbers of graphs 69 (2003) 229--239. |

[6] | P. Turán, On an extremal problem in graph theory, Math. Fiz. Lapok 48 (1941) 436--452. |

[7] | B. Zelinka, On signed , manuscript.2-independence numbers of graphs |

Received 21 February 2012

Revised 3 September 2012

Accepted 3 September 2012