Abstract
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.
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