MONOPHONIC DISTANCE IN GRAPHS

Abstract

For any two vertices u and v in a connected graph G, a u – v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) from u to v is defined as the length of a longest u – v monophonic path in G. A u – v monophonic path of length dm(u, v) is called a u – v monophonic. The monophonic eccentricity em(v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that rad m G ≤ diam m G for every connected graph G and that every pair a, b of positive integers with a ≤ b is realizable as the monophonic radius and monophonic diameter of some connected graph. Also, for any three positive integers a, b and c with 3 ≤ a ≤ b ≤ c, there is a connected graph G such that rad G = a, rad m G = b and rad DG = c; and for any three positive integers a, b and c with 5 ≤ a ≤ b ≤ c, there is a connected graph G such that diam G = a, diam m G = b and diam D G = c, where rad G, diam G, rad DG and diam D G denote the radius, diameter, detour radius and detour diameter, respectively. The monophonic center of G is the subgraph induced by the vertices of G having monophonic eccentricity rad m G and it is shown that every graph is the monophonic center of some connected graph and also that the monophonic center Cm(G) of every connected graph G is a subgraph of some block of G.