Fair domination in graphs

A fair dominating set in a graph G (or FD-set) is a dominating set S such that all vertices not in S are dominated by the same number of vertices from S ; that is, every two vertices not in S has the same number of neighbors in S . The fair domination number, fd ( G ) , of G is the minimum cardinality of an FD-set. Among other results, we show that if G is a connected graph of order n ≥ 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 G is a maximal outerplanar graph, then fd ( G ) < 17 n / 19 . If T is a tree of order n ≥ 2 , then we prove that fd ( T ) ≤ n / 2 with equality if and only if T is the corona of a tree.