The vertex monophonic number of a graph
A. P. Santhakumaran, P. Titus · Discussiones Mathematicae Graph Theory · 2012
For a connected graph G of order p ≥ 2 and a vertex x of G, a setWe determine bounds for it and characterize graphs which realize these bounds.A connected graph of order p with vertex monophonic numbers either p -1 or p -2 for every vertex is characterized.It is shown that for positive integers a, b and n ≥ 2 with 2 ≤ a ≤ b, there exists a connected graph G with rad m G = a, diam m G = b and m x (G) = n for some vertex x in G. Also, it is shown that for each triple m, n and p of integers with 1 ≤ n ≤ pm -1 and m ≥ 3, there is a connected graph G of order p, monophonic diameter m and m x (G) = n for some vertex x of G.