Total Roman Domination Number of Rooted Product Graphs

Abel Cabrera Martínez,Frank A Hernández Mira,Andrés Carrión García,Suitberto Cabrera García

doi:10.3390/math8101850

Abstract

Let G be a graph with no isolated vertex and f:V(G)→{0,1,2} a function. If f satisfies that every vertex in the set {v∈V(G):f(v)=0} is adjacent to at least one vertex in the set {v∈V(G):f(v)=2}, and if the subgraph induced by the set {v∈V(G):f(v)≥1} has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight ω(f)=∑v∈V(G)f(v) among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.

Highlights

The study of domination-related parameters in product graphs is one of the most important and attractive areas of domination theory in graphs
We study a very well-known variant of domination for the case of rooted product graphs
Given a graph G of order n and a graph H with root v ∈ V ( H ), the rooted product graph G ◦v H is defined as the graph obtained from G and H by taking one copy of G and n copies of H and identifying the ith-vertex of G with vertex v in the ith-copy of H for every i ∈ {1, . . . , n} [23]

Summary

Introduction

The study of domination-related parameters in product graphs is one of the most important and attractive areas of domination theory in graphs. The minimum weight among all TRDFs on G is the total Roman domination number, and is denoted γtR ( G ). The theorem, which we can consider as one of the main results of this paper, states the intervals in which the total Roman domination number of a rooted product graph can be found.

Results

Conclusion