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 γw(G) of (G,w) is the minimum weight w(D)=∑v∈Dw(v) of a set D⊆V(G) such that every vertex x∈V(D)−D has a neighbor in D. If ∑v∈V(G)w(v)=|V(G)|, then we speak of a normed weighted graph. Recently, we proved thatγw(G)γw(Ḡ)⩽n281+2n−2 and γw(G)+γw(Ḡ)⩽3n4+n2(n−2)for normed weighted bipartite graphs (G,w) of order n such that neither G nor the complement Ḡ has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs.