The Upper Triangle Free Detour Domination Number and the Forcing Triangle Free Detour Domination Number of a Graph
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Abstract
A minimum triangle free detour dominating set of G is a triangle free detour dominating set of S if no appropriate subset of S is a detour triangle dominating set of G. The upper triangle free detour domination number, indicated by γdnΔf+(G), is defined as the highest cardinality of a minimal triangle free detour domination set of G. This concept's general qualities are investigated. The upper triangle free detour domination number of a family of graphs is calculated. It is demonstrated that for every pair of positive integers 3 ≤ a ≤ b, there exists a connected graph G such that γdnΔf(G) = a andγΔfdn+(G)= b, where γΔfdn(G) is G's triangle free detour dominance number.This notion is investigated for several general features. A graph family's forced triangle free domination number can be computed. The maximum bipolar fuzzy spanning tree is used to introduce bipolar fuzzy detour g-interior and g-boundary nodes in a bipolar fuzzy tree. It is demonstrated that for every pair of positive integers a and b with 0≤ a ≤ b, there exists a connected graph G such that fγΔfdn(G) = a and, where γΔfdn(G)= bγΔfdn(G) is G's triangular free domination number.
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