Publications Repository - Gdańsk University of Technology

Imagine that we are given a set D of officials and a set W of civils. For each civil x ∈ W, there must be an official v ∈ D that can serve x, and whenever any such v is serving x, there must also be another civil w ∈ W that observes v, that is, w may act as a kind of witness, to avoid any abuse from v. 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 D of a graph G = (VG, EG) is said to be certified if every vertex in D has either zero or at least two neighbours in VG \ D. The cardinality of a minimum certified dominating set in G is called the certified domination number of G. 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.

Authors

• dr inż. Magda Dettlaff link open in new tab ,
• dr inż. Magdalena Lemańska link open in new tab ,
• Jerzy Topp,
• Radosław Ziemann,
• Paweł Żyliński