Upper double monophonic number of a graph.

Abstract

A set S of a connected graph G of order n is called a double monophonic set of G if for every pair of vertices x, y in G there exist vertices u, v in S such that x, y lie on a u - v monophonic path. The double monophonic number dm(G) of G is the minimum cardinality of a double monophonic set. A double monophonic set S in a connected graph G is called a minimal double monophonic set if no proper subset of S is a double monophonic set of G. The upper double monophonic number of G is the maximum cardinality of a minimal double monophonic set of G, and is denoted by dm+(G). Some general properties satisfied by upper double monophonic sets are discussed. It is proved  that for a connected graph G of order n, dm(G) = n if and only if dm+(G) = n. It is also proved that dm(G) = n - 1 if and only if dm+ (G) = n - 1 for a non-complete graph G of order n with a full degree vertex. For any positive integers 2 ≤ a ≤ b, there exists a connected graph G with dm(G) = a and dm+(G) = b.