Some new generalizations of Domination using restrictions on degrees of vertices

A set \(D\) of vertices in a graph \(G=(V,E)\) is a degree restricted dominating set for \(G\) if each vertex \(v_i\) in \(D\) is dominating atmost \(g(d_i)\) vertices of \(V-D\), where \(g\) is a function restricting the degree value \(d_i\) with respect to the given function value \(k_i\) for a natural valued function \(f\) from the vertex set of the graph. We define three different types of Degree Restricted Domination by varying the way how the restricted function \(g(v_i)\) is defined. If \(g(d_i)=\big\lceil \frac{d_i}{k_i}\big\rceil\), the corresponding domination is called the ceil degree restricted domination, in short, \(CDRD\), and the dominating set obtained in this manner is the \(CDRD\)-set. If \(g(d_i)=\big\lfloor\frac{d_i}{k_i}\big\rfloor\) or \(g(d_i)=d_i-k_i+1\), then the corresponding dominations are respectively called the floor degree restricted domination, in short \(FDRD\), or the translate degree restricted domination, \(TDRD\). The dominating sets obtained in this manner are the \(FDRD\)-set and the \(TDRD\)-set respectively. In this paper, we introduce these new generalizations of the domination number in line with the different \(DRD\)-sets and study these types of domination for some classes of graphs like complete graphs, caterpillar graphs etc. Degree restricted domination has a vital role in retaining the efficiency of nodes in a network and has many interesting applications.