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Perfect Domination, Roman Domination and Perfect Roman Domination in Lexicographic Product Graphs.

Cabrera Martínez Abel, García-Gómez César, Rodríguez-Velázquez Juan Alberto

Fundamenta Informaticae

2022

Vol. 185, nr 3

201--220

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

A note on the independent Roman domination in unicyclic graphs

Chellali M., Rad N. J.

Opuscula Mathematica

2012

Vol. 32, no. 4

715-718

A Roman dominating function (RDF) on a graph G= (V, E) is a function ƒ : V → {0, 1, 2} satisfying the condition that every vertex u for which ƒ(u) = 0 is adjacent to at least one vertex v for which ƒ(v)=2. The weight of an RDF is the value [formula]. An RDF ƒ in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number ΥR (G) (respectively, the independent Roman domination number ΥR(G) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that ΥR(G) strongly equals iR(G), denoted by ΥR(G) ≡ iR(G), if every RDF on G of minimum weight is independent. In this note we characterize all unicyclic graphs G with ΥR(G) ≡ iR(G).