@article{Titus_Eldin Vanaja_2017, title={Edge fixed monophonic number of a graph.}, volume={36}, url={https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/2381}, abstractNote={<p>For an edge <em>xy</em> in a connected graph <em>G</em> of order <em>p ≥ 3</em>, a set <em>S</em>C<em>V(G)</em>is an <em>xy</em>-monophonic set of <em>G</em> if each vertex <em>v Є V(G)</em> lies on an <em>x-u</em> monophonic path or a <em>y-u</em> monophonic path for some element <em>u</em> in <em>S</em>. The minimum cardinality of an <em>xy-</em> monophonic set of <em>G</em> is defined as the <em>xy-</em>monophonic number of <em>G</em>, denoted by <em>m<sub>xy</sub> (G)</em>. An <em>xy-</em>monophonic set of cardinality <em>m<sub>xy</sub> (G) </em>is called a <em>m<sub>xy</sub></em> -set of G. We determine bounds for it and find the same for special classes of graphs. It is shown that for any three positive integers <em>r, d</em> and <em>n </em><em>≥ 2 </em>with <em>2 ≤ r ≤ d</em>, there exists a connected graph <em>G</em> with monophonic radius <em>r</em>, monophonic diameter <em>d</em> and <em>m<sub>xy</sub> (G) = n</em> for some edge <em>xy</em> in <em>G</em>.</p>}, number={3}, journal={Proyecciones (Antofagasta, On line)}, author={Titus, P. and Eldin Vanaja, S.}, year={2017}, month={Oct.}, pages={363-372} }