On average lower independence and domination numbers in graphs (Q556830)

Let \(V(G)\) be the vertex set of a graph \(G\). The lower independence number \(i_{av}(G)\) of a graph \(G\) is defined as \(\frac{1}{| V(G)| }\sum_{v\in V(G)}i_v(G)\), and the average lower domination number \({\gamma}_{av}(G)\) is defined as \(\frac{1}{| V(G)| }\sum_{v\in V(G)}{\gamma}_v(G)\), where \(i_v(G)\) is the minimum cardinality of a maximal independent set containing \(v\), and \(\gamma_v(G)\) is the minimum cardinality of a dominating set containig \(v\). The authors present simple upper bounds for \(i_{av}(G)\) and \({\gamma}_{av}(G)\), and they characterize the graphs achieving these bounds.