The paper introduces a new soft graph by defining a set-valued function F. Consider \({G}^{*}=(V,E)\) as a simple graph and A as a minimal dominating set. Let R be a subset of \(A\times V\) , representing an arbitrary relation from \(A\) to \(V\) . A function \(F:A\to P(V)\) is defined as \(F\left(x\right)=\left\{x,y\in V|d(x,y)\le 2\right\}\) and a function \(K:A\to P\left(E\right)\) is defined as \(K\left(x\right)=\{xy\) or/and \(yz\in E|d\left(x,y\right)=1\) and \(d\left(x,z\right)=2\}\) , where \(P\) is a power set. The pair \((F,A)\) forms a soft set over \(V\) and \((K,A)\) forms a soft set over \(E\) . Then \((F\left(a\right),K\left(a\right))\) is a subgraph of \({G}^{*}, \forall a\in A\) . The paper investigates the properties of the soft graph based on the new parameter and presents a real-life application.
