Note on 2-rainbow domination and Roman domination in graphs

• Published in: Applied mathematics letters Vol. 23; no. 6; pp. 706 - 709
• Main Authors: Wu, Yunjian, Xing, Huaming
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.06.2010; Elsevier
• Subjects: 2-rainbow domination; Domination number; Exact sciences and technology; Graphs; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Proving; Roman; Roman domination; Sciences and techniques of general use; Unions; Upper bounds; Roman domination; Domination number; 2-rainbow domination; Lower bound; Upper bound; Applied mathematics; Connected graph
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2010.02.012

A Roman dominating function of a graph G is a function f : V → { 0 , 1 , 2 } such that every vertex with 0 has a neighbor with 2. The minimum of f ( V ( G ) ) = ∑ v ∈ V f ( v ) over all such functions is called the Roman domination number γ R ( G ) . A 2- rainbow dominating function of a graph G is a function g that assigns to each vertex a set of colors chosen from the set { 1 , 2 } , for each vertex v ∈ V ( G ) such that g ( v ) = 0̸ , we have ⋃ u ∈ N ( v ) g ( u ) = { 1 , 2 } . The 2- rainbow domination number γ r 2 ( G ) is the minimum of w ( g ) = ∑ v ∈ V | g ( v ) | over all such functions. We prove γ r 2 ( G ) ≤ γ R ( G ) and obtain sharp lower and upper bounds for γ r 2 ( G ) + γ r 2 ( G ¯ ) . We also show that for any connected graph G of order n ≥ 3 , γ r 2 ( G ) + γ ( G ) 2 ≤ n . Finally, we give a proof of the characterization of graphs with γ R ( G ) = γ ( G ) + k for 2 ≤ k ≤ γ ( G ) .