Secure Total Domination in Rooted Product Graphs

Abel Cabrera Martínez,Alejandro Estrada-Moreno,Juan A Rodríguez-Velázquez

doi:10.3390/math8040600

Abel Cabrera Martínez, Alejandro Estrada-Moreno + Show 1 more

Open Access

https://doi.org/10.3390/math8040600

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Journal: Mathematics	Publication Date: Apr 15, 2020
Citations: 2	License type: CC BY 4.0

Affiliation: Rovira i Virgili University

Abstract

In this article, we obtain general bounds and closed formulas for the secure total domination number of rooted product graphs. The results are expressed in terms of parameters of the factor graphs involved in the rooted product.

Highlights

Many authors have considered the following approach to the problem of protecting a graph [1,2,3,4,5,6,7]: suppose that one “entity” is stationed at some of the vertices of a graph G and that an entity at a vertex can deal with a problem at any vertex in its closed neighbourhood
The minimum cardinality among all secure total dominating sets of G is the secure total domination number of G, which is denoted by γst ( G )
The problem of computing γst ( G ) is NP-hard [18], even when restricted to chordal bipartite graphs, planar bipartite graphs with arbitrary large girth and maximum degree three, split graphs and graphs of separability at most two. This suggests finding the secure total domination number for special classes of graphs or obtaining tight bounds on this invariant. This is precisely the aim of this article in which we study the case of rooted product graphs

Summary

Introduction

Many authors have considered the following approach to the problem of protecting a graph [1,2,3,4,5,6,7]: suppose that one “entity” is stationed at some of the vertices of a (simple) graph G and that an entity at a vertex can deal with a problem at any vertex in its closed neighbourhood. The minimum cardinality among all secure total dominating sets of G is the secure total domination number of G, which is denoted by γst ( G ) This parameter was introduced by Benecke et al in [2] and studied further in [3,4,16,18,19]. The problem of computing γst ( G ) is NP-hard [18], even when restricted to chordal bipartite graphs, planar bipartite graphs with arbitrary large girth and maximum degree three, split graphs and graphs of separability at most two This suggests finding the secure total domination number for special classes of graphs or obtaining tight bounds on this invariant. This is precisely the aim of this article in which we study the case of rooted product graphs

Some Notation and Tools

The Case of Rooted Product Graphs

Concluding Remarks