Weighted domination in triangle-free graphs
The weighted domination number \(\gamma_w(G)\) of a weighted graph \((G,w)\), with positive weights on vertices, is the minimum weight of a dominating set of \(G\). A graph is normed if \(\sum_{v\in V(G)}=|V(G)|\). Earlier, the authors proved the following two Nordhaus-Gaddum type results \[ \gamma_w(G)\cdot\gamma_w(\overline G)\leq\frac{n^2}8 (1+\frac 2{n-2})\quad\text{and}\quad \gamma_w(G)+\gamma_w(\overline G)\leq\frac{3n}4+\frac n{2(n-2)} \] for normed weighted bipartite graphs \((G,w)\) of order \(n\) such that neither \(G\) nor \(\overline G\) has isolated vertices. In this paper, the authors extend these inequalities to normed triangle-free graphs, satisfying the same condition.
scientific article; zbMATH DE number 1792384