Discussiones Mathematicae Graph Theory 30(4) (2010) 563-574
doi: 10.7151/dmgt.1514

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Michael A. Henning

Department of Mathematics
University of Johannesburg
Auckland Park, 2006 South Africa
e-mail: mahenning@uj.ac.za

Christian Löwenstein and Dieter Rautenbach

Institut für Mathematik
TU Ilmenau, Postfach 100565
D-98684 Ilmenau, Germany
e-mails: {christian.loewenstein, dieter.rautenbach}@tu-ilmenau.de


A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph.

Keywords: domination, total domination, domatic number, vertex partition, Petersen graph.

2010 Mathematics Subject Classification: 05C69.


[1] N.J. Calkin and P. Dankelmann, The domatic number of regular graphs, Ars Combin. 73 (2004) 247-255.
[2] G.S. Domke, J.E. Dunbar and L.R. Markus, The inverse domination number of a graph, Ars Combin. 72 (2004) 149-160.
[3]U. Feige, M.M. Halldórsson, G. Kortsarz and A. Srinivasan, Approximating the domatic number, SIAM J. Comput. 32 (2002) 172-195, doi: 10.1137/S0097539700380754.
[4] C. Godsil and G. Royle, Algebraic Graph Theory (Springer, 2001).
[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in graphs: Advanced topics (Marcel Dekker, New York, 1998).
[7] S.M. Hedetniemi, S.T. Hedetniemi, R.C. Laskar, L. Markus and P.J. Slater, Disjoint dominating sets in graphs, in: Proc. Internat. Conf. Discrete Math., ICDM 2006, 87-100, Ramanujan Math. Soc., Lecture Notes Series in Mathematics, 2008.
[8] 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.
[9] M.A. Henning and J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159-162.
[10] M.A. Henning and J. Southey, A characterization of graphs with disjoint dominating and total dominating sets, Quaestiones Mathematicae 32 (2009) 119-129, doi: 10.2989/QM.2009.
[11] V.R. Kulli and S.C. Sigarkanti, Inverse domination in graphs, Nat. Acad. Sci. Lett. 14 (1991) 473-475.
[12] C. Löwenstein and D. Rautenbach, Pairs of disjoint dominating sets and the minimum degree of graphs, Graphs Combin. 26 (2010) 407-424, doi: 10.1007/s00373-010-0918-9.
[13] O. Ore, Theory of Graphs, Amer. Math. Soc. Transl. 38 (Amer. Math. Soc., Providence, RI, 1962) 206-212.
[14] B. Zelinka, Total domatic number and degrees of vertices of a graph, Math. Slovaca 39 (1989) 7-11.
[15] B. Zelinka, Domatic numbers of graphs and their variants: A survey, in: Domination in graphs: Advanced topics, T.W. Haynes et al. eds (Marcel Dekker, New York, 1998), 351-377.

Received 18 July 2009
Revised 9 November 2009
Accepted 9 November 2009