Certified domination

TL;DR

This paper introduces the concept of certified domination in graphs, modeling a security scenario with officials and witnesses, and explores its properties, bounds, and computational complexity.

Contribution

It defines the certified domination number, provides exact values for certain graphs, bounds for general graphs, and proves the NP-completeness of the decision problem.

Findings

Exact certified domination numbers for specific graph classes

Upper bounds for arbitrary graphs

NP-completeness of the certified domination problem

Abstract

Imagine that we are given a set $D$ of officials and a set $W$ of civils. For each civil $x \in W$, there must be an official $v \in D$ that can serve $x$, and whenever any such $v$ is serving $x$, there must also be another civil $w \in W$ 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 perfectly models the aforementioned problem. Specifically, a dominating set $D$ of a graph $G=(V_G,E_G)$ is said to be certified if every vertex in $D$ has either zero or at least two neighbours in $V_G\setminus D$. 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. Finally, we show that the (decision) certified domination problem is NP-complete.