Connected Fair Domination in Graphs

Abstract

In this paper, we introduce the notion of connected fair domination in graphs. A connected fair dominating set in a graph G (or $\mathsf {CFD}$-set) is a dominating set S such that $\langle S \rangle$ is connected in G and all vertices not in S are dominated by the same number of vertices from S, i.e., every two vertices not in S has the same number of neighbours in S. The connected fair domination number of G ($\mathsf {cfd}(G)$) is the minimum cardinality of a $\mathsf {CFD}$-set in G. Apart from finding $\mathsf {cfd}(G)$ for some standard graphs G, we proved various bounds on $\mathsf {cfd}(G)$ in terms of order and some other graph parameters of G.