Upper Detour Monophonic Number of a Graph

For a connected graph G=(V,E)G=(V,E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path   if it is a chordless path. A longest x−yx−y monophonic path is called an x−yx−ydetour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G   lies on an x−yx−y detour monophonic path for some elements x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G   and is denoted by dm(G)dm(G). A detour monophonic set S of G is called a minimal detour monophonic set if no proper subset of S is a detour monophonic set of G. The upper detour monophonic number of G  , denoted by dm+(G)dm+(G), is defined as the maximum cardinality of a minimal detour monophonic set of G  . We determine bounds for it and find the upper detour monophonic number of certain classes of graphs. It is shown that for any three positive integers a,b,ca,b,c with 2≤a≤b≤c2≤a≤b≤c, there is a connected graph G   with m(G)=a,dm(G)=bm(G)=a,dm(G)=b and dm+(G)=cdm+(G)=c. Also, for any three positive integers a,ba,b and n   with 2≤a≤n≤b2≤a≤n≤b, there is a connected graph G   with dm(G)=a,dm+(G)=bdm(G)=a,dm+(G)=b and a minimal detour monophonic set of cardinality n.