Disjunctive domination in graphs with minimum degree at least two (Q6041864)

From the author's abstract: ``A set \(D\) of vertices in a graph \(G\) is called a disjunctive dominating set in \(G\) if every vertex not in \(D\) is adjacent to a vertex in \(D\) or has at least two vertices in \(D\) at distance \(2\) from it in \(G\). The disjunctive domination number, denoted \(\gamma^d_2(G)\), of \(G\) is the minimum cardinality of a disjunctive dominating set in \(G\).'' In this paper, the author proves that if \(G\) is a graph of order \(n \geq3\) with \(\delta(G) \geq 2\) such that no component in \(G\) is isomorphic to any of eight specific graphs given in the paper, then \(\gamma^d_2(G) \leq \frac{n}{3}\). The author also provides an infinite family of graphs that attain the upper bound of \(\frac{n}{3}\).