DMGT

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

PARTITIONING A GRAPH INTO A DOMINATING SET, A TOTAL DOMINATING SET, AND SOMETHING ELSE

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

Abstract

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.

References

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