On the connected detour monophonic number of a graph

Abstract

For a connected graph G = (V, E) of order at least two, a connected detour monophonic set S of G is called a minimal connected detour monophonic set if no proper subset of S is a connected detour monophonic set of G. The upper connected detour monophonic number of G, denoted by dm+c (G), is defined as the maximum cardinality of a minimal connected detour monophonic set of G. We determine bounds for it and find the same for some special classes of graphs. For any three positive integers a, b and n with 6 ≤ a ≤ n ≤ b, there is a connected graph G with dmc(G) = a, dm+c (G) = b and a minimal connected detour monophonic set of cardinality n.