Abstract
For a graph G, the x-triangle free detour set, the x-triangle free detour number, the minimal x-triangle free detour set, the upper x-triangle free detour number, are defined and studied. Certain bounds are determined and the relation with the vertex triangle free detour number of a graph is found out. Some realization problems, properties related to the upper vertex detour number, the upper vertex detour monophonic number and the upper vertex geodetic number are also studied.
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