Restrained Differential of a Graph

Abstract

Let $G = (V, E)$ be an arbitrary graph. For any subset X of V, let B(X) be the set of all vertices in $V-X$ that have a neighbour in X. Mashburn et al. defined the differential of a set X to be $\partial (X) = |B(X)| - |X|$, and the differential of a graph is max$\{\partial (X)\}$, where the maximum is taken over all subsets X of V. Motivated by this parameter we define the restrained differential of graph as follows. For any subset X of V, let $\overline{B}(X)$ be the set of all vertices in $V-X$ that have a neighbor in X and a neighbour in $V-X$. We define the restrained differential of a set X to be $\overline{\partial }(X) = |\overline{B}(X)| - |X|$ and the restrained differential of a graph is max$\{\overline{\partial }(X)\}$, where the maximum is taken over all subsets X of V. In this paper, we initiate a study of this parameter.