THE UPPER VERTEX MONOPHONIC NUMBER OF A GRAPH
P. Titus, K. Iyappan · International Journal of Pure and Apllied Mathematics · 2016
For any vertex x in a connected graph G of order p ≥ 2, a setx (G), is defined as the maximum cardinality of a minimal x-monophonic set of G.We determine bounds for it and find the same for some special classes of graphs.For any two positive integers a and b with 1 ≤ a ≤ b, there exists a connected graph G with mx(G) = a and m +x (G) = b for some vertex x in G. Also, it is shown that for any three positive integers a, b and n with a ≥ 2 and a ≤ n ≤ b, there exists a connected graph G with mx(G) = a, m +x (G) = b and a minimal x-monophonic set of cardinality n.