Some new generalizations of Domination using restrictions on degrees of vertices

Abstract

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.