The Upper Connected Vertex Detour Monophonic Number of a Graph

For any vertex x in a connected graph G of order n ≥ 2, a set S x ⊆ V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S x such that the subgraph induced by S x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm x (G). A connected x-detour monophonic set S x of G is called a minimal connected x-detour monophonic set if no proper subset of S x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm+ x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm x (G) = j,dm+ x (G) = k and cdm+ x (G) = l for some vertex x in G.