A note of 2-rainbow edge domination numbers of graphs

Abstract

Let G be a graph with edge set E containing no isolated vertices. A 2-rainbow edge dominating function of G is a function f from E to the family of all subsets of {1, 2} such that for each edge e ∈ E with f (e) = ᶲ, where . The minimum value of a 2-rainbow edge dominating function f of G is called the 2-rainbow edge domination number of G, denoted by . A Roman {2}-edge dominating function of G is 0, 1, 2} such that for each edge e ∈ E with g(e) = 0. The minimum value of a Roman {2}-edge dominating function g of G is called the Roman {2}-edge domination number of G, denoted by . An edge dominating set of G is a set F ⊆ E such that each edge not in F is adjacent to an edge in F . The edge domination number of G is the minimum cardinality among all edge dominating sets of G. In this paper, we first prove that for any tree T . Secondly, for any tree T , we present a lower bound on (T ) in terms of and characterize the trees T for which . Finally, it is known that for any graph G, and we characterize the trees T for which .