Upper Detour Monophonic Number of a Graph

Abstract

For a connected graph 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−y monophonic path is called an x−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−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). 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), 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,c with 2≤a≤b≤c, there is a connected graph G with m(G)=a,dm(G)=b and dm+(G)=c. Also, for any three positive integers a,b and n with 2≤a≤n≤b, there is a connected graph G with dm(G)=a,dm+(G)=b and a minimal detour monophonic set of cardinality n.