An edge of a graph is called dot-critical if its contraction decreases the domination number. A graph is said to be dot-critical if all of its edges are dot-critical. A vertex of a graph is called critical if its deletion decreases the domination number. In A note on the domination dot-critical graphs, Discrete Appl. Math. 157 (2009) 3743-3745, Chen and Shiu constructed for each even integer k ≥ 4 infinitely many k-dot-critical graphs G with no critical vertices and k(G) = 1. In this paper, we refine their result and construct for integers k ≥ 4 and l ≥ 1 infinitely many k-dot-critical graphs G with no critical vertices, k(G) = 1 and λ(G) = l. Furthermore, we prove that every 3-dot- critical graph with no critical vertices is 3-connected, and it is best possible.
@article{bwmeta1.element.doi-10_7151_dmgt_1752,
author = {Michitaka Furuya},
title = {The Connectivity Of Domination Dot-Critical Graphs With No Critical Vertices},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {683-690},
zbl = {1303.05136},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1752}
}
Michitaka Furuya. The Connectivity Of Domination Dot-Critical Graphs With No Critical Vertices. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 683-690. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1752/
[1] T. Burton and D.P. Sumner, Domination dot-critical graphs, Discrete Math. 306 (2006) 11-18. doi:10.1016/j.disc.2005.06.029 | Zbl 1085.05047
[2] X. Chen and W.C. Shiu, A note on the domination dot-critical graphs, Discrete Appl. Math. 157 (2009) 3743-3745. doi:10.1016/j.dam.2009.07.014 | Zbl 1227.05204
[3] R. Diestel, Graph Theory 4th Edition (Verlag, Heidelberg, Springer, 2010).
[4] M. Furuya and M. Takatou, Upper bound on the diameter of a domination dot- critical graph, Graphs Combin. 29 (2013) 79-85. doi:10.1007/s00373-011-1095-1 | Zbl 1258.05093
[5] D.A. Mojdeh and S. Mirzamani, On the diameter of dot-critical graphs, Opuscula Math. 29 (2009) 165-175. | Zbl 1204.05070
[6] N.J. Rad, On the diameter of a domination dot-critical graph, Discrete Appl. Math. 157 (2009) 1647-1649. doi:10.1016/j.dam.2008.10.015 | Zbl 1182.05066