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Triple Roman Domination in Graphs

Hossein Abdollahzadeh Ahangar, M. Pilar Alvarez, Mustapha Chellali, Seyed Mahmoud Sheikholeslami, Juan Carlos Valenzuela-Tripodoro

TL;DR

This manuscript starts the study of a variant of Roman domination in graphs: the triple Roman domination, which considers that any city of the Roman Empire must be able to be defended by at least three legions.

Abstract

The Roman domination in graphs is well-studied in graph theory. The topic is related to a defensive strategy problem in which the Roman legions are settled in some secure cities of the Roman Empire. The deployment of the legions around the Empire is designed in such a way that a sudden attack to any undefended city could be quelled by a legion from a strong neighbour. There is an additional condition: no legion can move if doing so leaves its base city defenceless. In this manuscript we start the study of a variant of Roman domination in graphs: the triple Roman domination. We consider that any city of the Roman Empire must be able to be defended by at least three legions. These legions should be either in the attacked city or in one of its neighbours. We determine various bounds on the triple Roman domination number for general graphs, and we give exact values for some graph families. Moreover, complexity results are also obtained.

Triple Roman Domination in Graphs

TL;DR

This manuscript starts the study of a variant of Roman domination in graphs: the triple Roman domination, which considers that any city of the Roman Empire must be able to be defended by at least three legions.

Abstract

The Roman domination in graphs is well-studied in graph theory. The topic is related to a defensive strategy problem in which the Roman legions are settled in some secure cities of the Roman Empire. The deployment of the legions around the Empire is designed in such a way that a sudden attack to any undefended city could be quelled by a legion from a strong neighbour. There is an additional condition: no legion can move if doing so leaves its base city defenceless. In this manuscript we start the study of a variant of Roman domination in graphs: the triple Roman domination. We consider that any city of the Roman Empire must be able to be defended by at least three legions. These legions should be either in the attacked city or in one of its neighbours. We determine various bounds on the triple Roman domination number for general graphs, and we give exact values for some graph families. Moreover, complexity results are also obtained.
Paper Structure (5 sections, 23 theorems, 41 equations, 4 figures)

This paper contains 5 sections, 23 theorems, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Complexity results
  3. Bounds in terms of $p,\Delta$ and $\delta$
  4. Relationships with some Roman domination parameters
  5. TRD-number in some families of graphs.

Key Result

Theorem 1

Problem TRIPLE ROM-DOM is NP-Complete for bipartite graphs.

Figures (4)

  • Figure 1: Increasing the defence up to three times does not always increase the cost to triple.
  • Figure 2: NP-completeness of 3RDF for bipartite.
  • Figure 3: The condition $g(\Gamma)\ge 5$ is necessary in Proposition \ref{['cota2']} .
  • Figure 4: 3RDF s for $C_{4},C_{5},C_{7},C_{10}$

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Proposition 7
  • Corollary 8
  • Proposition 9
  • Corollary 10
  • Proposition 11
  • ...and 13 more