The main goal of this paper is to compare the proof of the existence and uniqueness of fixed points for $\mathcal{T}$-contractive mappings in various metric spaces and different distances regarding some techniques in mathematical analysis. Also, several comparisons to show the efficiency of the obtained result will be considered.‎ Manuscript profile

‎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, ‎$i(\mathcal{H})=\omega(\mathcal{H})$. Manuscript profile