# The Connectivity Of Domination Dot-Critical Graphs With No Critical Vertices

• Volume: 34, Issue: 4, page 683-690
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Michitaka Furuya. "The Connectivity Of Domination Dot-Critical Graphs With No Critical Vertices." Discussiones Mathematicae Graph Theory 34.4 (2014): 683-690. <http://eudml.org/doc/269821>.

@article{MichitakaFuruya2014,
abstract = {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.},
author = {Michitaka Furuya},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {dot-critical graph; critical vertex; connectivity.; connectivity},
language = {eng},
number = {4},
pages = {683-690},
title = {The Connectivity Of Domination Dot-Critical Graphs With No Critical Vertices},
url = {http://eudml.org/doc/269821},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Michitaka Furuya
TI - The Connectivity Of Domination Dot-Critical Graphs With No Critical Vertices
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 4
SP - 683
EP - 690
AB - 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.
LA - eng
KW - dot-critical graph; critical vertex; connectivity.; connectivity
UR - http://eudml.org/doc/269821
ER -

## References

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1. [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 Zbl1085.05047
2. [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 Zbl1227.05204
3. [3] R. Diestel, Graph Theory 4th Edition (Verlag, Heidelberg, Springer, 2010).
4. [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 Zbl1258.05093
5. [5] D.A. Mojdeh and S. Mirzamani, On the diameter of dot-critical graphs, Opuscula Math. 29 (2009) 165-175. Zbl1204.05070
6. [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 Zbl1182.05066

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