Restrained domination in unicyclic graphs

Abstract

Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then γ r (U) ≥ n 3 , and provide a characterization of graphs achieving this bound.

Determination of the Restrained Domination Number on Vertex Amalgamation and Edge Amalgamation of the Path Graph With the Same Order

“…𝛾 𝑟 (𝐺) ≥ 𝑛 − 2𝑚 3 [16] (2) However, before presenting the theorem and its proof, it is first shown some definitions and observations from the development of the concept of the restrained dominating set and the amalgamation of the vertices and edges of the path graph that will be used in proving the theorem. ⌋ always above or equal to the lower limit of the restrained domination number of a graph ∎…”

Himpunan Dominasi Terkendali Graf Hasil Operasi Amalgamasi Titik Dan Sisi Pada Graf Siklus Berorde Sama

“…Penelitian tentang himpunan dominasi terkendali juga telah banyak dikaji, diantaranya "Restrained Domination in Graphs" yang memperoleh hasil bilangan dominasi terkendali pada graf lengkap ( (( )), graf lintasan ( (' )), dan graf siklus ( ( )) [5]. Sedangkan "On equality in an upper bound for the restrained and total domination numbers of a graph" [6] dan "Restrained Domination in Unicyclic Graphs" [7] masing -masing memperoleh hasil batas atas dan batas bawah bilangan dominasi terkendali dari suatu graf. Selain itu, penelitian tentang himpunan dominasi terkendali yang melibatkan operasi graf dari beberapa graf khusus juga telah banyak dikaji, diantaranya "Restrained Domination in Graphs Under Some Binary Operations" [8], "The Product of the Restrained Domination Numbers of A Graph and Its Complement" [9], "Secure Restrained Domination in the Join and Corona of Graphs" [10], "Penentuan Bilangan Dominasi Terkendali pada ' ⨀' dan ( ⨀( " [11], dan "Determination of the Restrained Domination Number on Vertex Amalgamation and Edge Amalgamation of the Path Graph With the Same Order" [12].…”

“…Adapun teorema yang akan digunakan untuk pembuktian hasil penelitian ini adalah: ( ) = − 2 , untuk ≥ 3 [5] dan pembuktian batas atas dan batas bawah bilangan dominasi terkendali dengan masing -masing menggunakan teorema ( ) , − ∆, dengan . ≥ 2 [6] dan ( ) ≥ − / , untuk order dan size (7).…”

On domination multisubdivision number of unicyclic graphs

“…A family of unicyclic graphs is widely studied by many authors in the theory of domination, see for example [7,10,11].…”