Gated independence in graphs (Q6546421)

An independent set \(I\) of a graph \(G\) is gated if for each vertex \(u\in S\) there exists its neighbor \(u'\) such that \((I\setminus \{u\})\cup \{u'\}\) is an independent set in \(G\). The gated independence number \(\operatorname{gi}(G)\) of \(G\) is the cardinality of the largest gated independent set in \(G\). The paper presents a number of lower and upper bounds on \(\operatorname{gi}(G)\) involving invariants of \(G\) such as the induced matching number, the uniquely restricted matching number, the independence domination number, the paired domination numbers, as well as the order, the size, and the maximum degree. In particular, if \(G\) is a subcubic graph with no isolated vertex and no component isomorphic to \(K_{3,3}\), then \(\operatorname{gi}(G)\) is at least one-fifth of the order of \(G\). Graphs in which every independent set is gated are also considered and numerous problems are posed.