Global Domination in Fuzzy Graph Using Valid Edges

In the realm of a crisp graph G = (V, E), where V is the vertex set and E is the edges set, a fuzzy graph G^∗ = (σ, µ) defined over V consists of two functions: σ : V → [0, 1] and µ : E → [0, 1]. For each pair of vertices u, v ∈ V, the membership degree of the edge µ(uv) is constrained by σ(u) ∧ σ(v), where ∧ denote minimum operator. The validity of edges in G^∗ is determined by the membership value of each edge; specifically, an edge xy ∈ E is considered a valid edge if its validity I ≥ 0.5. The objective of this study is to present the notion of a global dominating set based on valid edges. A subset D ⊆ V is recognized as a dominating set in the fuzzy graph G^∗ if there is a vertex v ∈ D such that v dominates every u ∈ V \D, implying the validity of the edge vu. Moreover, if the set D acts as the dominating set in both the fuzzy graph complement G^∗ and the fuzzy graph G^∗, it is termed the global dominating set. The global domination number, denoted as γ_g, signifies the minimum fuzzy cardinality of the set D. This research employs an axiomatic deductive approach, starting with a set of axioms and then utilizing deductive logic to advance arguments related to the studied topic. Subsequently, the obtained results yield various properties related to dominating sets and global dominating sets on fuzzy graphs based on their valid edges. We limit our consideration to a fuzzy graph that is undirected and lacks any loops or multiple edges.