Characterization of Fractional Mixed Domination Number of Paths and Cycles

Let G′ be a simple, connected, and undirected (UD) graph with the vertex set M(G′) and an edge set N(G′). In this article, we define a function as a fractional mixed dominating function (FMXDF) if it satisfies for all , where indicates the closed mixed neighbourhood of , that is the set of all such that is adjacent to and is incident with and also itself. Here, is the poundage (or weight) of f. The fractional mixed domination number (FMXDN) is denoted by and is designated as the lowest poundage among all FMXDFs of . We compute the FMXDN of some common graphs such as paths, cycles, and star graphs, the middle graph of paths and cycles, and shadow graphs. Furthermore, we compute upper bounds for the sum of the two fractional dominating parameters, resulting in the inequality , where and are the fractional edge domination number and FMXDN, respectively. Finally, we compare to other resolvability-related parameters such as metric and fault-tolerant metric dimensions on some families of graphs.