Discussiones Mathematicae Graph Theory 33(1) (2013)
139-146

doi: 10.7151/dmgt.1652

*This paper is dedicated to Mietek Borowiecki on the occasion of his 70th
birthday. We thank him for his warmth and kindness to us, for his inspiration,
and for his dedication to graph theory.*

Michael A. Henning
Department of Mathematics | Douglas F. Rall
Department of Mathematics |

**Keywords:** domination, *2*-domination, vertex partition

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

[1] | M. Dorfling, W. Goddard, M.A. Henning and C.M. Mynhardt, Construction of trees and graphs with equal domination parameters, Discrete Math. 306 (2006) 2647--2654, doi: 10.1016/j.disc.2006.04.031. |

[2] | S.M. Hedetniemi, S.T. Hedetniemi, R.C. Laskar, L. Markus and P.J. Slater, Disjoint dominating sets in graphs, Proc. ICDM 2006, Ramanujan Mathematics Society Lecture Notes Series 7 (2008) 87--100. |

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

[4] | M.A. Henning, C. Löwenstein and D. Rautenbach, Remarks about disjoint dominating sets, Discrete Math. 309 (2009) 6451--6458, doi: 10.1016/j.disc.2009.06.017. |

[5] | M.A. Henning, C. Löwenstein and D. Rautenbach, An independent dominating set in the complement of a minimum dominating set of a tree, Appl. Math. Lett. 23 (2010) 79--81, doi: 10.1016/j.aml.2009.08.008. |

[6] | M.A. Henning, C. Löwenstein and D. Rautenbach, Partitioning a graph into a dominating set, a total dominating set, and something else, Discuss. Math. Graph Theory 30 (2010) 563--574, doi: 10.7151/dmgt.1514. |

[7] | M.A. Henning and J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159--162. |

[8] | M.A. Henning and J. Southey, A characterization of graphs with disjoint dominating and total dominating sets, Quaest. Math. 32 (2009) 119--129. |

[9] | O. Ore, Theory of Graphs: Amer. Math. Soc. Colloq. Publ., 38 (Amer. Math. Soc., Providence, RI, 1962). |

[10] | J. Southey and M.A. Henning, Dominating and total dominating partitions in cubic graphs, Central European J. Math. 9(3) (2011) 699--708, doi: 10.2478/s11533-011-0014-2. |

[11] | J. Southey and M.A. Henning, A characterization of graphs with disjoint dominating and paired-dominating sets, J. Comb. Optim. 22 (2011) 217--234, doi: 10.1007/s10878-009-9274-1. |

[12] | B. Zelinka, Total domatic number and degrees of vertices of a graph, Math. Slovaca 39 (1989) 7--11. |

Received 28 February 2012

Revised 2 October 2012

Accepted 31 October 2012