Abstract
A vertex of a graph is called critical if its deletion decreases the domination number, and an edge 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. In this paper, we show that if G is a connected dot-critical graph with domination number k ≥ 3 and diameter d and if G has no critical vertices, then d ≤ 2k−3.
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References
Burton T., Sumner D.P.: Domination dot-critical graphs. Discret. Math. 306, 11–18 (2006)
Chengye Z., Yuansheng Y., Linlin S.: Domination dot-critical graphs with no critical vertices. Discret. Math. 308, 3241–3248 (2008)
Diestel R.: Graph Theory, 4th edn. Graduate Texts in Mathematics, vol. 173. Springer, Berlin (2010)
Mojdeh D.A., Mirzamani S.: On the diameter of dot-critical graphs. Opuscula Math. 29, 165–175 (2009)
Rad N.J.: On the diameter of a domination dot-critical graph. Discret. Appl. Math. 157, 1647–1649 (2009)
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Furuya, M., Takatou, M. Upper Bound on the Diameter of a Domination Dot-Critical Graph. Graphs and Combinatorics 29, 79–85 (2013). https://doi.org/10.1007/s00373-011-1095-1
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DOI: https://doi.org/10.1007/s00373-011-1095-1