Fractional Double Domination in Graphs and Dual Graphs

Objective: This study provides the fractional double domination number of a graph and analyses its essential aspects. Method: A fractional double dominating function of a graph 𝐺 = (𝑉 , 𝐸) is defined as a function 𝑓 ∶ 𝑉 → [0, 1] such that Σ𝑢∈𝑁[𝑣] 𝑓(𝑢) ≥ 2 for every vertex 𝑣 ∈ 𝑉 , where 𝑁[𝑣] denotes the closed neighborhood of 𝑣. The fractional double domination number, denoted by 𝛾𝑓𝑑(𝐺), is the minimum weight Σ𝑣∈𝑉 𝑓(𝑣) taken over all such functions. Findings: This study establishes general bounds for 𝛾𝑓𝑑(𝐺) and derives exact values for several standard classes of graphs including paths, cycles, stars, wheels, and complete graphs. Furthermore, the analysis is extended to dual graphs, presenting new bounds and relationships between the fractional double domination numbers of a graph and its dual. Illustrative examples and constructive assignments are provided to demonstrate the tightness of the bounds. Novelty: The results presented herein not only broaden the scope of fractional domination theory but also contribute to the growing body of research on domination parameters with redundancy and robustness in networks.