Let γₜ(G) and γ₂(G) be the total domination number and the 2-domination number of a graph G, respectively. It has been shown that: γₜ(T) ≤ γ₂(T) for any tree T. In this paper, we provide a constructive characterization of those trees with equal total domination number and 2-domination number.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1504, author = {You Lu and Xinmin Hou and Jun-Ming Xu and Ning Li}, title = {A characterization of (gt,g2)-trees}, journal = {Discussiones Mathematicae Graph Theory}, volume = {30}, year = {2010}, pages = {425-435}, zbl = {1217.05181}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1504} }
You Lu; Xinmin Hou; Jun-Ming Xu; Ning Li. A characterization of (γₜ,γ₂)-trees. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 425-435. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1504/
[000] [1] M. Blidia, M. Chellalia and T.W. Haynes, Characterizations of trees with equal paired and double domination numbers, Discrete Math. 306 (2006) 1840-1845, doi: 10.1016/j.disc.2006.03.061. | Zbl 1100.05068
[001] [2] M. Blidia, M. Chellali and L. Volkmann, Some bounds on the p-domination number in trees, Discrete Math. 306 (2006) 2031-2037, doi: 10.1016/j.disc.2006.04.010. | Zbl 1100.05069
[002] [3] E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219, doi: 10.1002/net.3230100304. | Zbl 0447.05039
[003] [4] E.J. Cockayne, O. Favaron, C.M. Mynhardt and J. Puech, A characterization of (γ,i)-trees, J. Graph Theory 34 (2000) 277-292, doi: 10.1002/1097-0118(200008)34:4<277::AID-JGT4>3.0.CO;2-# | Zbl 0949.05059
[004] [5] G. Chartrant and L. Lesniak, Graphs & Digraphs, third ed. (Chapman & Hall, London, 1996).
[005] [6] J.F. Fink and M.S. Jacobson, n-Domination in graphs, in: Y. Alavi, A.J. Schwenk (eds.), Graph Theory with Applications to Algorithms and Computer Science (Wiley, New York, 1985) 283-300. | Zbl 0573.05049
[006] [7] F. Harary and M. Livingston, Characterization of trees with equal domination and independent domination numbers, Congr. Numer. 55 (1986) 121-150. | Zbl 0647.05020
[007] [8] T.W. Haynes, S.T. Hedetniemi, M.A. Henning and P.J. Slater, H-forming sets in graphs, Discrete Math. 262 (2003) 159-169, doi: 10.1016/S0012-365X(02)00496-X. | Zbl 1017.05082
[008] [9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (New York, Marcel Deliker, 1998). | Zbl 0890.05002
[009] [10] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (New York, Marcel Deliker, 1998). | Zbl 0883.00011
[010] [11] T.W. Haynes, M.A. Henning and P.J. Slater, Strong quality of domination parameters in trees, Discrete Math. 260 (2003) 77-87, doi: 10.1016/S0012-365X(02)00451-X. | Zbl 1020.05051
[011] [12] M.A. Henning, A survey of selected recently results on total domination in graphs, Discrete Math. 309 (2009) 32-63, doi: 10.1016/j.disc.2007.12.044.
[012] [13] X. Hou, A characterization of (2γ,γₚ)-trees, Discrete Math. 308 (2008) 3420-3426, doi: 10.1016/j.disc.2007.06.034. | Zbl 1165.05023