A New Protection Model : the Eternal Feedback Vertex Problem

Un nouveau modèle de protection de graphes : problème de l'ensemble de rebouclement éternel

• Auteur / Autrice : Nour Dyab
• Direction : Hamamache Kheddouci
• Type : Thèse de doctorat
• Discipline(s) : Informatique
• Date : Soutenance le 24/10/2023
• Etablissement(s) : Lyon 1
• Ecole(s) doctorale(s) : École doctorale InfoMaths (Lyon ; 2009-....)
• Partenaire(s) de recherche : Laboratoire : Laboratoire d'InfoRmatique en Images et Systèmes d'information (Ecully, Rhône ; 2003-....)
• Jury : Président / Présidente : Salima Benbernou
• Jury : Examinateurs / Examinatrices : Hamamache Kheddouci, Olivier Togni, Abdallah Makhoul, Noura Faci, Mohammed Lalou, Maidoun Mortada
• Jury : Rapporteur / Rapporteuse : Olivier Togni, Abdallah Makhoul

Mots clés

Mots clés contrôlés

Protection de l'information (informatique)

Logique informatique

Protection

Arbres (théorie des graphes) -- Informatique

Algorithmes

Mots clés libres

Eternel

Modèle de protection

Algorithme

Eternal

Algorithm

Feedback Vertex set

Guard

Graph protection models

Résumé

Cette thèse vise à développer un modèle de protection de graphe en utilisant des gardes mobiles, en se concentrant spécifiquement sur le concept d'ensembles de sommets de rétroaction éternelle.

This thesis aims to develop a model for graph protection using mobile guards, specifically focusing on the concept of eternal feedback vertex sets. The problem of protecting a graph using mobile guards has been extensively studied in the literature. This problem involves defending the vertices, as well as edges, of a graph G against any attack using defense units, called guards, stationed at the vertices of G. Our primary interest lies in protecting graphs against infinite sequences of attacks, where each attack is executed one at a time. A protection problem can be modeled by a two-player game between a defender and an attacker. In this game, the defender selects the initial configuration of guards in the first turn and must defend all vertices of G against any attack by reconfiguring the guards at each subsequent turn. On the other hand, the attacker chooses the location of each attack at each turn. An attack is considered defended if a guard can be moved to the attacked vertex via an edge. The defender wins the game if they successfully defend the graph against every sequence of attacks, while the attacker wins if he manages to breach the defense. Importantly, the sequence of attacks can be of infinite length. Variants of this problem, such as eternal dominating set, eternal independent set, and eternal covering have been well explored in the literature. In this thesis, we introduce two new variants of the graph protection problem, namely Eternal Feedback Vertex Sets (EFVS ) and m-Eternal Feedback Vertex Sets (m-EFVS ). The two variants are modeled in the same way using a two-player game, with the specific condition that the set of vertices chosen by the defender to receive guards must simultaneously form a feedback vertex set and a dominating set in each turn. In other words, the Eternal Feedback Vertex Set on a graph can be seen as an infinite game between a defender and an attacker, where the defender chooses a set of guards Fi at each turn i, i ≥ 1. At turn i, the attacker chooses a vertex ri, called an attack, in V \ Fi−1 and the defender must defend the attack by moving to ri a guard from a vertex vj adjacent to ri. The new guards configuration is Fi = Fi−1 ∪ {ri} \ {vj }. The sets Fi, for any i ≥ 1, must be dominating and feedback vertex sets. In the case where the defender protects vertices by moving more than one guard, we talk about the m-Eternal Feedback Vertex Sets.