Certified domination

Abstract

Imagine that we are given a set of officials and a set of civils. For each civil , there must be an official that can serve , and whenever any such is serving , there must also be another civil that observes , that is, may act as a kind of witness, to avoid any abuse from . What is the minimum number of officials to guarantee such a service, assuming a given social network? In this paper, we introduce the concept of certified domination that models the aforementioned problem. Specifically, a dominating set of a graph is said to be certified if every vertex in has either zero or at least two neighbours in . The cardinality of a minimum certified dominating set in is called the certified domination number of . Herein, we present the exact values of the certified domination number for some classes of graphs as well as provide some upper bounds on this parameter for arbitrary graphs. We then characterise a wide class of graphs with equal domination and certified domination numbers and characterise graphs with large values of certified domination numbers. Next, we examine the effects on the certified domination number when the graph is modified by deleting/adding an edge or a vertex. We also provide Nordhaus–Gaddum type inequalities for the certified domination number.

Keywords

• certified domination
• domination
• corona
• nordhaus–gaddum