Abstract
A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph induced by V∖S is connected. The total outer-connected domination number γ toc (G) is the minimum size of such a set. We give some properties and bounds for γ toc in general graphs and in trees. For graphs of order n, diameter 2 and minimum degree at least 3, we show that \(\gamma_{toc}(G)\le \frac{2n-2}{3}\) and we determine the extremal graphs.
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References
Brigham RC, Carrington JR, Vitray RP (2000) Connected graphs with maximal total domination number. J Comb Math Comb Comput 34:81–95
Cockayne EJ, Dawes RM, Hedetniemi ST (1980) Total domination in graphs. Networks 10:211–219
Cyman J (2007) The outer-connected domination number of a graph. Australas J Comb 38:35–46
Cyman J (2010) Total outer-connected domination numbers in trees. Discuss Math, Graph Theory 30:377–383
Cyman J, Lemańska M, Raczek J (2006) On the doubly connected domination number of a graph. Cent Eur J Math 4:34–45
Cyman J, Raczek J (2009) Total outer-connected domination numbers of trees. Discrete Appl Math 157:3198–3202
Hattingh JH, Joubert EJ (2010) A note on total outer-connected domination numbers of a tree. AKCE J Graphs Comb 7:223–227
Acknowledgement
This research was in part supported by a grant from IPM (No.90050043)
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Favaron, O., Karami, H. & Sheikholeslami, S.M. On the total outer-connected domination in graphs. J Comb Optim 27, 451–461 (2014). https://doi.org/10.1007/s10878-012-9531-6
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DOI: https://doi.org/10.1007/s10878-012-9531-6