Characterization of Fractional Mixed Domination Number of Paths and Cycles

Abstract

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 $f:M\cup N⟶\left[0,1\right]$ as a fractional mixed dominating function (FMXDF) if it satisfies $f\left(R_m\left[x\right]\right)={{\displaystyle \sum }}_{yϵR_m\left[x\right]}f\left(y\right)\ge 1$ for all $x\in M\left(G^{\prime }\right)\cup N\left(G^{\prime }\right)$, where $R_m\left[x\right]$ indicates the closed mixed neighbourhood of $x$, that is the set of all $y\in M\left(G^{\prime }\right)\cup N\left(G^{\prime }\right)$ such that $y$ is adjacent to $x$ and $y$ is incident with $x$ and also $x$ itself. Here, $p\left(f\right)={{\displaystyle \sum }}_{x\in M\cup N}f\left(x\right)$ is the poundage (or weight) of f. The fractional mixed domination number (FMXDN) is denoted by ${\gamma }_{fm}^{\ast }\left(G^{\prime }\right)$ and is designated as the lowest poundage among all FMXDFs of $G^{\prime }$. 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 ${\gamma }_{f_1}^{\prime }\left({\rm T}\right)+{\gamma }_{fm}^{\ast }\left({\rm T}\right)\le r+p-\text{rad}\left({\rm T}\right)-\alpha$, where ${\gamma }_{f_1}^{\prime }$ and ${\gamma }_{fm}^{\ast }$ are the fractional edge domination number and FMXDN, respectively. Finally, we compare ${\gamma }_{fm}^{\ast }$ to other resolvability-related parameters such as metric and fault-tolerant metric dimensions on some families of graphs.