The vertex detour hull number of a graph

For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval I_D[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), ${\displaystyle I_D\left[S\right]={\bigcup }_{x,y\in S}{I}_D[x,y]}$. A set S of vertices is a detour convex set if ${\displaystyle I_D\left[S\right]=S}$. The detour convex hull ${\displaystyle {\left[S\right]}_D}$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of V(G) with ${\displaystyle {\left[S\right]}_D=V\left(G\right)}$. Let x be any vertex in a connected graph G. For a vertex y in G, denoted by ${\displaystyle I_D{\left[y\right]}^x}$, the set of all vertices distinct from x that lie on some x - y detour of G; while for S ⊆ V(G), ${\displaystyle I_D{\left[S\right]}^x={\bigcup }_{y\in S}{I}_D{\left[y\right]}^x}$. For x ∉ S, S is an x-detour convex set if ${\displaystyle I_D{\left[S\right]}^x=S}$. The x-detour convex hull of S, ${\displaystyle {\left[S\right]}^x\_D}$ is the smallest x-detour convex set containing S. A set S is an x-detour hull set if ${\displaystyle {\left[S\right]}^x\_D=V\left(G\right)-\left\{x\right\}}$ and the minimum cardinality of x-detour hull sets is the x-detour hull number dhₓ(G) of G. For x ∉ S, S is an x-detour set of G if ${\displaystyle I_D{\left[S\right]}^x=V\left(G\right)-\left\{x\right\}}$ and the minimum cardinality of x-detour sets is the x-detour number dₓ(G) of G. Certain general properties of the x-detour hull number of a graph are studied. It is shown that for each pair of positive integers a,b with 2 ≤ a ≤ b+1, there exist a connected graph G and a vertex x such that dh(G) = a and dhₓ(G) = b. It is proved that every two integers a and b with 1 ≤ a ≤ b, are realizable as the x-detour hull number and the x-detour number respectively. Also, it is shown that for integers a,b and n with 1 ≤ a ≤ n -b and b ≥ 3, there exist a connected graph G of order n and a vertex x such that dhₓ(G) = a and the detour eccentricity of x, ${\displaystyle e_D\left(x\right)=b}$. We determine bounds for dhₓ(G) and characterize graphs G which realize these bounds.