The Upper Monophonic Number of a Graph

Abstract

For a connected graph G = (V,E), a Smarandachely k-monophonic set of G is a set M ⊆ V (G) such that every vertex of G is contained in a path with less or equal k chords joining some pair of vertices in M. The Smarandachely k-monophonic number m k(G) of G is the minimum order of its Smarandachely k-monophonic sets. Particularly, a Smarandachely 0-monophonic path, a Smarandachely 0-monophonic number is abbreviated to a monophonic path, monophonic number m(G) of G respectively. Any monophonic set of order m(G) is a minimum monophonic set of G. A monophonic set M in a connected graph G is called a minimal monophonic set if no proper subset of M is a monophonic set of G. The upper monophonic number m + (G) of G is the maximum cardinality of a minimal monophonic set of G. Connected graphs of order p with upper monophonic number p and p − 1 are characterized. It is shown that for every two integers a and b such that 2 ≤ a ≤ b, there exists a connected graph G with m(G) = a and m + (G) = b.