1. bookAHEAD OF PRINT
Journal Details
License
Format
Journal
eISSN
2083-5892
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English
access type Open Access

Minimal Graphs with Disjoint Dominating and Total Dominating Sets

Published Online: 08 Oct 2021
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 23 Feb 2021
Accepted: 19 Sep 2021
Journal Details
License
Format
Journal
eISSN
2083-5892
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English
Abstract

A graph G is a DTDP-graph if it has a pair (D, T) of disjoint sets of vertices of G such that D is a dominating set and T is a total dominating set of G. Such graphs were studied in a number of research papers. In this paper we study further properties of DTDP-graphs and, in particular, we characterize minimal DTDP-graphs without loops.

Keywords

MSC 2010

[1] I. Broere, M. Dorfling, W. Goddard, J.H. Hattingh, M.A. Henning and E. Ungerer, Augmenting trees to have two disjoint total dominating sets, Bull. Inst. Combin. Appl. 42 (2004) 12–18.Search in Google Scholar

[2] P. Delgado, W.J. Desormeaux and T.W. Haynes, Partitioning the vertices of a graph into two total dominating sets, Quaest. Math. 39 (2016) 863–873. https://doi.org/10.2989/16073606.2016.118886210.2989/16073606.2016.1188862Search in Google Scholar

[3] W.J. Desormeaux, T.W. Haynes and M.A. Henning, Partitioning the vertices of a cubic graph into two total dominating sets, Discrete Appl. Math. 223 (2017) 52–63. https://doi.org/10.1016/j.dam.2017.01.03210.1016/j.dam.2017.01.032Search in Google Scholar

[4] M. Dorfling, W. Goddard, J.H. Hattingh and M.A. Henning, Augmenting a graph of minimum degree 2 to have two disjoint total dominating sets, Discrete Math. 300 (2005) 82–90. https://doi.org/10.1016/j.disc.2005.06.02010.1016/j.disc.2005.06.020Search in Google Scholar

[5] T.W. Haynes, S.T. Hedetniemi and M.A. Henning (Eds), Topics in Domination in Graphs (Dev. Math. 64, Springer, Cham, 2020). https://doi.org/10.1007/978-3-030-51117-310.1007/978-3-030-51117-3Search in Google Scholar

[6] T.W. Haynes, S.T. Hedetniemi and M.A. Henning (Eds), Structures of Domination in Graphs (Dev. Math. 66, Springer, Cham, 2021). https://doi.org/10.1007/978-3-030-58892-210.1007/978-3-030-58892-2Search in Google Scholar

[7] P. Heggernes and J.A. Telle, Partitioning graphs into generalized dominating sets, Nordic J. Comput. 5 (1998) 128–142.Search in Google Scholar

[8] 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. https://doi.org/10.7151/dmgt.151410.7151/dmgt.1514Search in Google Scholar

[9] M.A. Henning, C. Löwenstein, D. Rautenbach and J. Southey, Disjoint dominating and total dominating sets in graphs, Discrete Appl. Math. 158 (2010) 1615–1623. https://doi.org/10.1016/j.dam.2010.06.00410.1016/j.dam.2010.06.004Search in Google Scholar

[10] M.A. Henning and A.J. Marcon, Semitotal domination in graphs: Partition and algorithmic results, Util. Math. 106 (2018) 165–184.Search in Google Scholar

[11] M.A. Henning and I. Peterin, A characterization of graphs with disjoint total dominating sets, Ars Math. Contemp. 16 (2019) 359–375. https://doi.org/10.26493/1855-3974.1525.7f310.26493/1855-3974.1525.7f3Search in Google Scholar

[12] M.A. Henning and J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159–162.Search in Google Scholar

[13] M.A. Henning and J. Southey, A characterization of graphs with disjoint dominating and total dominating sets, Quaest. Math. 32 (2009) 119–129. https://doi.org/10.2989/QM.2009.32.1.10.71210.2989/QM.2009.32.1.10.712Search in Google Scholar

[14] M.A. Henning and A. Yeo, Total Domination in Graphs (Springer Monogr. Math., Springer, New York, 2013). https://doi.org/10.1007/978-1-4614-6525-610.1007/978-1-4614-6525-6Search in Google Scholar

[15] E.M. Kiunisala and F.P. Jamil, On pairs of disjoint dominating sets in a graph, Int. J. Math. Anal. 10 (2016) 623–637. https://doi.org/10.12988/ijma.2016.634310.12988/ijma.2016.6343Search in Google Scholar

[16] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, Amer. Math. Soc., Providence, 1962).Search in Google Scholar

[17] J. Southey and M.A. Henning, Dominating and total dominating partitions in cubic graphs, Cent. Eur. J. Math. 9 (2011) 699–708. https://doi.org/10.2478/s11533-011-0014-210.2478/s11533-011-0014-2Search in Google Scholar

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