Domination numbers and diameters in certain graphs
محورهای موضوعی : Combinatorics, Graph theory
S. S. Karimizad
1
(Department of Mathematics, Faculty of Basic Sciences, Ilam University, Ilam, Iran)
کلید واژه: Domination number, bicritical graph, diameter.,
چکیده مقاله :
Regarding the problem mentioned by Brigham et al. ``Is it correct that each connected bicritical graph possesses a minimum dominating set having every two appointed vertices of graphs?", we first give a class of graphs that disprove it and second obtain domination numbers and diameters of the graphs of this class. This class of graphs has the property: $\omega(\mathcal{H}) - diam(\mathcal{H})\rightarrow \infty$ when $|\mathcal{V}(\mathcal{H})|= n \rightarrow \infty$. Also, for the bicritical graphs of this class, $i(\mathcal{H})=\omega(\mathcal{H})$.
Regarding the problem mentioned by Brigham et al. ``Is it correct that each connected bicritical graph possesses a minimum dominating set having every two appointed vertices of graphs?", we first give a class of graphs that disprove it and second obtain domination numbers and diameters of the graphs of this class. This class of graphs has the property: $\omega(\mathcal{H}) - diam(\mathcal{H})\rightarrow \infty$ when $|\mathcal{V}(\mathcal{H})|= n \rightarrow \infty$. Also, for the bicritical graphs of this class, $i(\mathcal{H})=\omega(\mathcal{H})$.
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