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

Discussiones Mathematicae Graph Theory (2014)

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

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topMichitaka 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

top- [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] 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] 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 Zbl1258.05093
- [5] D.A. Mojdeh and S. Mirzamani, On the diameter of dot-critical graphs, Opuscula Math. 29 (2009) 165-175. Zbl1204.05070
- [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 Zbl1182.05066

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