A note on total domination and 2-rainbow domination in graphs

Abstract

Let G be a graph without isolated vertices. A total dominating set of G is a subset S of V(G) such that every vertex of G is adjacent to a vertex in S. The minimum cardinality of a total dominating set of G is denoted by γt(G). A 2-rainbow dominating function of G is a function f:V(G)→2{1,2} such that for each v∈V(G) with f(v)=0̸, ⋃u∈NG(v)f(u)={1,2}. The minimum of ∑v∈V(G)|f(v)| over all 2-rainbow dominating functions f of G is denoted by γr2(G).Chellali, Haynes and Hedetniemi conjectured that for every graph G without isolated vertices, γt(G)≤γr2(G). In this note, we solve the conjecture.