The Upper Total Triangle Free Detour Number of a Graph

: For a connected graph G = (V,E) of order at least two, a total triangle free detour set of a graph G is a triangle free detour set S such that the subgraph G[S] induced by S has no isolated vertices. The minimum cardinality of a total triangle free detour set of G is the total triangle free detour number of G. It is denoted by 〖 tdn 〗 _Δf (G). A total triangle free detour set of cardinality 〖 tdn 〗 _Δf (G) is called 〖 tdn 〗 _Δf-set of G. In this article, the concept of upper total triangle free detour number of a graph G is introduced. It is found that the upper total triangle free detour number differs from total triangle free detour number. The upper total triangle free detour number is found for some standard graphs. Their bounds are determined. Certain general properties satisfied by them are studied.


Introduction
For a graph  = (, ), we mean a finite undirected connected simple graph.The order of G is represented by n.We consider graphs with at least two vertices.For basic definitions we refer [3].For vertices u and v in a connected graph , the detour distance (, ) is the length of the longest  −  path in G.A  - path of length  (, ) is called a  −  detour.This concept was studied by Chartrand et.al [1].
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 − y detour 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 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).The detour monophonic number of a graph was introduced in [8] and further studied in [7].
A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph G[S] induced by S has no isolated vertices.The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G).A total detour monophonic set of cardinality dmt(G) is called a dmt-set of G.These concepts were studied by A. P. Santhakumaran et.al [6].
The concept of triangle free detour distance was introduced by Keerthi Asir and Athisayanathan [4].A path P is called a triangle free path if no three vertices of P induce a triangle.For vertices u and v in a connected graph G, the triangle free detour distance DΔf (u, v) is the length of a longest u − v triangle free path in G.A uv path of length DΔf (u, v) is called a u − v triangle free detour.For any two vertices u and v in a connected graph The triangle free detour eccentricity of a vertex  in a connected graph  is defined by  ∆ () = max{ ∆ (, ): ,  ∈ } .The triangle free detour radius of is defined by  ∆ () = min{ ∆ ():  ∈ } and The triangle free detour diameter of is defined by  ∆ () = max { ∆ ():  ∈ } A total triangle free detour set of a graph G is a triangle free detour set S such that the subgraph G[S] induced by S has no isolated vertices.The minimum cardinality of a total triangle free detour set of G is the total triangle free detour number of G.It is denoted by tdn Δf (G).A total triangle free detour set of cardinality tdn Δf (G) is called tdn Δf -set of G.
A vertex v of a connected graph G is called a support vertex of G if it is adjacent to an end vertex of G. Two adjacent vertices are referred to as neighbors of each other.The set N(v) of neighbors of a vertex v is called the neighborhood of v.A vertex v of a graph G is called extreme vertex if the subgraph induced by its neighbourhood is complete.The following theorems will be used in the sequel.Theorem 1.1: Let G be a connected graph of order n, then 2 ≤ dn Δf (G) ≤ tdn Δf (G) ≤ cdn Δf (G) ≤ n.