Characterization of Fractional Mixed Domination Number of Paths and Cycles

초록

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 f : M boolean OR N -> 0,1 as a fractional mixed dominating function (FMXDF) if it satisfies f (R-m [x]) = Sigma(y epsilon Rm[x]) fy >= 1 for all x is an element of MG 'boolean OR NG ', where R-m[x] indicates the closed mixed neighbourhood of x, that is the set of all y is an element of MG 'boolean OR NG ' such that y is adjacent to x and y is incident with x and also x itself. Here, p(f) = Sigma(x is an element of M boolean OR N)f(x) is the poundage (or weight) of f. The fractional mixed domination number (FMXDN) is denoted by gamma(fm)*(G ') and is designated as the lowest poundage among all FMXDFs of G '. 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 gamma(f1)'(T)+gamma(fm)*(T) <= r + p - rad(T) - alpha, where gamma(f1)' and gamma(fm)* are the fractional edge domination number and FMXDN, respectively. Finally, we compare gamma(fm)* to other resolvability-related parameters such as metric and fault-tolerant metric dimensions on some families of graphs.

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