Hyperbolicity on Graph Operators

Abstract

A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) ≤ δ ( R ( G ) ) ≤ δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) ≤ δ ( Q ( G ) ) ≤ δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) ≤ 2 δ ( R ( G ) ) ≤ δ ( S ( G ) ) + 1 and δ ( R ( G ) ) − 1 / 2 ≤ δ ( Λ ( G ) ) ≤ 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.

Alianzas defensivas fuertes en gráficas clásicas bajo el operador S(G) y R(G)

“…También se han estudiado otros parámetros en este tipo de operadores como se evidencia en los trabajos de Castro (2023) y Bermúdez (2018. Motivados por investigaciones previas y en línea con el trabajo de donde se estudió el número de alianza defensiva en el operador línea.…”

Total Domination on Tree Operators

“…Motivated by the above works and by [12,13], where the total domination number is studied for some graph operators acting on general graphs, we study here the total domination number of several graph operators acting on trees. Other parameters have also been studied in this type of graphs (see [1,2,9]).…”

Total Domination on Some Graph Operators

“…Motivated by the previous investigations, we work here on the total domination number of some graph operators. Other important parameters have also been studied in this type of graphs (see References [7,8]).…”