The Global Connected Domination in Graphs

Abstract

A subset \(S\) of vertices of a graph \(G\) is called a global connected dominating set if \(S\) is both a global dominating set and a connected dominating set. The global connected domination number, denoted by \(\gamma_{gc}(G)\), is the minimum cardinality of a global connected dominating set of \(G\). In this paper, sharp bounds for \(\gamma_{gc}\) are supplied, and all graphs attaining those bounds are characterized. We also characterize all graphs of order \(n\) with \(\gamma_{gc} = k\), where \(3 \leq k \leq n-1\). Exact values of this number for trees and cycles are presented as well.