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

• Autorzy: Michitaka Furuya
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

dot-critical graph; critical vertex; connectivity.

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2014
• Tom: 34
• Numer: 4
• Strony: 683-690

Daty

wydano

2014-11-01

otrzymano

2013-07-08

poprawiono

2013-09-20

zaakceptowano

2013-09-30

online

2014-11-15

Twórcy

autor

Michitaka Furuya

• Department of Mathematical Information Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo

Bibliografia

• [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
• [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
• [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
• [5] D.A. Mojdeh and S. Mirzamani, On the diameter of dot-critical graphs, Opuscula Math. 29 (2009) 165-175.
• [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

Identyfikatory

DOI

10.7151/dmgt.1752