Weighted domination in triangle-free graphs

A weighted graph (G,w) is a graph G together with a positive weight-function on its vertex set w : V(G)→R^>0. The weighted domination number γ_w(G) of (G,w) is the minimum weight w(D)=∑_vDw(v) of a set DV(G) such that every vertex xV(D)−D has a neighbor in D. If ∑_vV(G)w(v)=|V(G)|, then we speak of a normed weighted graph. Recently, we proved that

for normed weighted bipartite graphs (G,w) of order n such that neither G nor the complement has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs.