Fair domination in graphs

A fair dominating set in a graph GG (or FD-set) is a dominating set SS such that all vertices not in SS are dominated by the same number of vertices from SS; that is, every two vertices not in SS has the same number of neighbors in SS. The fair domination number, fd(G), of GG is the minimum cardinality of an FD-set. Among other results, we show that if GG is a connected graph of order n≥3n≥3 with no isolated vertex, then fd(G)≤n−2, and we construct an infinite family of connected graphs achieving equality in this bound. We show that if GG is a maximal outerplanar graph, then fd(G)<17n/19. If TT is a tree of order n≥2n≥2, then we prove that fd(T)≤n/2 with equality if and only if TT is the corona of a tree.