Upper Edge Detour Monophonic Number of a Graph
For a connected graph G of order at least two, a path P is called a monophonic path if it is a chordless path. A longest x — y monophonic path is called an x — y detour monophonic path. A set S of vertices of G is an edge detour monophonic set of G if every edge of G lies on a detour monophonic path joining some pair of vertices in S.The edge detour monophonic number of G is the minimum cardinality of its edge detour monophonic sets and is denoted by edm(G).An edge detour monophonic set S ofG is called a minimal edge detour mono-phonic set ifno proper subset ofS is an edge detour monophonic set of G. The upper edge detour monophonic number of G, denoted by edm+(G),is defined as the maximum cardinality of a minimal edge detour monophonic set ofG. We determine bounds for it and characterize graphs which realize these bounds. For any three positive integers b, c and n with 2 ≤ b ≤ n ≤ c, there is a connected graph G with edm(G) = b, edm+(G) = c and a minimal edge detour monophonic set of cardinality n.