A self-stabilizing algorithm for optimally efficient sets in graphs

The efficiency of a set S⊆V in a graph G=(V,E), is defined as ε(S)=|{v∈V−S:|N(v)∩S|=1}|; in other words, the efficiency of a set S equals the number of vertices in V−S that are adjacent to exactly one vertex in S. A set S is called optimally efficient if for every vertex v∈V−S, ε(S∪{v})⩽ε(S), and for every vertex u∈S, ε(S−{u})